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(squares and cubes (non)dist)
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=== Sums of like powers===
+
==A055277 Triangle T(n,k) of number of rooted trees with n nodes and k leaves, 1 <= k <= n.==
==== Sums of k m-th powers >= 0 (Table A)====
+
The triangle starts:
{| class="wikitable" style="text-align:left"
+
      A002 A550 A550 A550 A550 A550 A550 A550 A550 A550 A550 
!   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!! !! !!m=13
+
        620  278  279  280  281  282  283  284  285  286  287
|-
+
n/k 1    2   3   4   5   6   7   8   9   10   11  12
| k&gt;=2 ||<span title="Sums of at least 2 squares s'', for s &gt;= 4.">[https://oeis.org/A176209 A176209]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
--+---------------------------------------------------------------
|-
+
  1| 1
| k=-1 ||<span title="Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.">[https://oeis.org/A336448 A336448]</span>||<span title="Sum of the cubes of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294287 A294287]</span>||<span title="Sum of the fourth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294288 A294288]</span>||<span title="Sum of the fifth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294300 A294300]</span>||<span title="Sum of the sixth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294301 A294301]</span>||<span title="Sum of the seventh powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294302 A294302]</span>||<span title="Numbers that are sums of 8th powers of 2 distinct positive integers.">[https://oeis.org/A155468 A155468]</span>||<span title="Sum of 9th powers.">[https://oeis.org/A007487 A007487]</span>||<span title="Sum of the tenth powers of the parts in the partitions of n into two distinct parts.">[https://oeis.org/A294305 A294305]</span>||&#xa0;||&#xa0;||<span title="Sum of 13th powers: 0^13+1^13+2^13+...+n^13.">[https://oeis.org/A181134 A181134]</span>
+
  2| 1    0
|-
+
  3| 1    1    0
| k=2 ||<span title="Sum of two squares of Lucas numbers (A000032).">[https://oeis.org/A140328 A140328]</span>||<span title="Sums of two nonnegative cubes.">[https://oeis.org/A004999 A004999]</span>||<span title="Numbers that are the sum of two 4th powers in more than one way.">[https://oeis.org/A018786 A018786]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
  4| 1    2    1    0
|-
+
  5| 1    4    3    1    0
| k=3 ||<span title="Numbers that are the sum of three squares (square 0 allowed) in exactly ten ways.">[https://oeis.org/A294713 A294713]</span>||<span title="Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| &gt;= |y| &gt;= |z|, or 0 if no such x exists.">[https://oeis.org/A332201 A332201]</span>||<span title="Numbers that are the sum of three biquadrates (fourth powers) in more than one way.">[https://oeis.org/A193244 A193244]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
  6| 1    6    8    4    1    0
|-
+
  7| 1    9  18  14    5    1    0
| k=4 ||&#xa0;||<span title="Numbers that are the sum of 4 cubes in more than 1 way.">[https://oeis.org/A001245 A001245]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
  8| 1  12  35  39  21    6    1    0
|-
+
  9| 1  16  62  97  72  30    7    1    0
|}
+
10| 1  20  103  212  214  120  40    8    1    0
==== Sums of exactly k positive m-th powers &gt; 0 (Table B)====
+
11| 1   25  161  429  563  416  185  52    9    1    0
{| class="wikitable" style="text-align:left"
+
12| 1  30  241  804 1344 1268  732  270  65  10    1    0
! &#xa0; !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
+
13| 1  36  348 1427 2958 3499 2544 1203  378  80  11    1    0
|-
+
For some columns, an o.g.f. is already given or conjectured:
| k=2 ||<span title="Numbers that are the sum of 2 nonzero squares, including repetitions.">[https://oeis.org/A024509 A024509]</span>||<span title="Numbers that are the sum of 2 positive cubes.">[https://oeis.org/A003325 A003325]</span>||<span title="Numbers that are the sum of 2 positive 4th powers.">[https://oeis.org/A003336 A003336]</span>||<span title="Numbers that are the sum of 2 positive 5th powers.">[https://oeis.org/A003347 A003347]</span>||<span title="Numbers that are the sum of 2 nonzero 6th powers.">[https://oeis.org/A003358 A003358]</span>||<span title="Numbers that are the sum of 2 positive 7th powers.">[https://oeis.org/A003369 A003369]</span>||<span title="Numbers that are the sum of 2 nonzero 8th powers.">[https://oeis.org/A003380 A003380]</span>||<span title="Numbers that are the sum of 2 positive 9th powers.">[https://oeis.org/A003391 A003391]</span>||<span title="Numbers that are the sum of 2 nonzero 10th powers.">[https://oeis.org/A004802 A004802]</span>||<span title="Numbers that are the sum of 2 positive 11th powers.">[https://oeis.org/A004813 A004813]</span>
+
A002620 (column 2): x^2 / ((1+x)*(1-x)^3)
|-
+
A055278 (column 3): x^4*(x^3+x^2+1) / ((1-x^2)*(1-x^3)*(1-x)^3)
| k=3 ||<span title="Numbers that are the sum of 3 nonzero squares, including repetitions.">[https://oeis.org/A024795 A024795]</span>||<span title="Numbers that are the sum of 3 positive cubes, including repetitions.">[https://oeis.org/A024981 A024981]</span>||<span title="Numbers that are the sum of 3 nonzero 4th powers in more than one way.">[https://oeis.org/A309762 A309762]</span>||<span title="Numbers that are the sum of 3 positive 5th powers.">[https://oeis.org/A003348 A003348]</span>||<span title="Numbers that are the sum of 3 nonzero 6th powers.">[https://oeis.org/A003359 A003359]</span>||<span title="Numbers that are the sum of 3 positive 7th powers.">[https://oeis.org/A003370 A003370]</span>||<span title="Numbers that are the sum of 3 nonzero 8th powers.">[https://oeis.org/A003381 A003381]</span>||<span title="Numbers that are the sum of 3 positive 9th powers.">[https://oeis.org/A003392 A003392]</span>||<span title="Numbers that are the sum of 3 nonzero 10th powers.">[https://oeis.org/A004803 A004803]</span>||<span title="Numbers that are the sum of 3 positive 11th powers.">[https://oeis.org/A004814 A004814]</span>
+
A055279 (column 4): x^5*(1+x+3*x^2+5*x^3+7*x^4+5*x^5+5*x^6+2*x^7+x^8) / ((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4))
|-
+
When o.g.f.s for higher columns are "guessed" (with Maple's <code>guessgf</code>), it turns out that the denominators are all products of cyclotomic polynomials. The following factorizations of the denominators are obtained:
| k=4 ||<span title="Numbers that are the sum of 4 nonzero squares.">[https://oeis.org/A000414 A000414]</span>||&#xa0;||<span title="Numbers that are the sum of 4 nonzero 4th powers in more than one way.">[https://oeis.org/A309763 A309763]</span>||<span title="Numbers that are the sum of 4 positive 5th powers.">[https://oeis.org/A003349 A003349]</span>||<span title="Numbers that are the sum of 4 positive 6th powers.">[https://oeis.org/A003360 A003360]</span>||<span title="Numbers that are the sum of 4 positive 7th powers.">[https://oeis.org/A003371 A003371]</span>||<span title="Numbers that are the sum of 4 nonzero 8th powers.">[https://oeis.org/A003382 A003382]</span>||<span title="Numbers that are the sum of 4 positive 9th powers.">[https://oeis.org/A003393 A003393]</span>||<span title="Numbers that are the sum of 4 nonzero 10th powers.">[https://oeis.org/A004804 A004804]</span>||<span title="Numbers that are the sum of 4 positive 11th powers.">[https://oeis.org/A004815 A004815]</span>
+
deo[2 ] :=                                                                                             (x^2-1)  *(x-1)^2;
|-
+
deo[3 ] :=                                                                                  (x^3-1)  *(x^2-1)  *(x-1)^3;
| k=5 ||<span title="Numbers that are the sum of 5 positive squares.">[https://oeis.org/A047700 A047700]</span>||<span title="Numbers that are the sum of 5 positive cubes.">[https://oeis.org/A003328 A003328]</span>||<span title="Numbers that are the sum of 5 positive 4th powers.">[https://oeis.org/A003339 A003339]</span>||<span title="Numbers that are the sum of 5 positive 5th powers.">[https://oeis.org/A003350 A003350]</span>||<span title="Numbers that are the sum of 5 positive 6th powers.">[https://oeis.org/A003361 A003361]</span>||<span title="Numbers that are the sum of 5 positive 7th powers.">[https://oeis.org/A003372 A003372]</span>||<span title="Numbers that are the sum of 5 nonzero 8th powers.">[https://oeis.org/A003383 A003383]</span>||<span title="Numbers that are the sum of 5 positive 9th powers.">[https://oeis.org/A003394 A003394]</span>||<span title="Numbers that are the sum of 5 positive 10th powers.">[https://oeis.org/A004805 A004805]</span>||<span title="Numbers that are the sum of 5 positive 11th powers.">[https://oeis.org/A004816 A004816]</span>
+
deo[4 ] :=                                                                        (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^3;
|-
+
deo[5 ] :=                                                              (x^5-1)  *(x^4-1)            *(x^2-1)^2*(x-1)^5;
| k=6 ||&#xa0;||<span title="Numbers that are the sum of 6 positive cubes.">[https://oeis.org/A003329 A003329]</span>||<span title="Numbers that are the sum of 6 positive 4th powers.">[https://oeis.org/A003340 A003340]</span>||<span title="Numbers that are the sum of 6 positive 5th powers.">[https://oeis.org/A003351 A003351]</span>||<span title="Numbers that are the sum of 6 positive 6th powers.">[https://oeis.org/A003362 A003362]</span>||<span title="Numbers that are the sum of 6 positive 7th powers.">[https://oeis.org/A003373 A003373]</span>||<span title="Numbers that are the sum of 6 nonzero 8th powers.">[https://oeis.org/A003384 A003384]</span>||<span title="Numbers that are the sum of 6 positive 9th powers.">[https://oeis.org/A003395 A003395]</span>||<span title="Numbers that are the sum of 6 positive 10th powers.">[https://oeis.org/A004806 A004806]</span>||<span title="Numbers that are the sum of 6 positive 11th powers.">[https://oeis.org/A004817 A004817]</span>
+
deo[6 ] :=                                                     (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^3;
|-
+
deo[7 ] :=                                            (x^7-1)*(x^6-1)  *(x^5-1) *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^4;
| k=7 ||&#xa0;||<span title="Numbers that are the sum of 7 positive cubes.">[https://oeis.org/A003330 A003330]</span>||<span title="Numbers that are the sum of 7 positive 4th powers.">[https://oeis.org/A003341 A003341]</span>||<span title="Numbers that are the sum of 7 positive 5th powers.">[https://oeis.org/A003352 A003352]</span>||<span title="Numbers that are the sum of 7 positive 6th powers.">[https://oeis.org/A003363 A003363]</span>||<span title="Numbers that are the sum of 7 positive 7th powers.">[https://oeis.org/A003374 A003374]</span>||<span title="Numbers that are the sum of 7 nonzero 8th powers.">[https://oeis.org/A003385 A003385]</span>||<span title="Numbers that are the sum of 7 positive 9th powers.">[https://oeis.org/A003396 A003396]</span>||<span title="Numbers that are the sum of 7 positive 10th powers.">[https://oeis.org/A004807 A004807]</span>||<span title="Numbers that are the sum of 7 positive 11th powers.">[https://oeis.org/A004818 A004818]</span>
+
deo[8 ] :=                                     (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)  *(x^2-1)^3*(x-1)^5;
|-
+
deo[9 ] :=                            (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^3*(x-1)^4;
| k=8 ||&#xa0;||<span title="Numbers that are the sum of 8 positive cubes.">[https://oeis.org/A003331 A003331]</span>||<span title="Numbers that are the sum of 8 positive 4th powers.">[https://oeis.org/A003342 A003342]</span>||<span title="Numbers that are the sum of 8 positive 5th powers.">[https://oeis.org/A003353 A003353]</span>||<span title="Numbers that are the sum of 8 positive 6th powers.">[https://oeis.org/A003364 A003364]</span>||<span title="Numbers that are the sum of 8 positive 7th powers.">[https://oeis.org/A003375 A003375]</span>||<span title="Numbers that are the sum of 8 nonzero 8th powers.">[https://oeis.org/A003386 A003386]</span>||<span title="Numbers that are the sum of 8 positive 9th powers.">[https://oeis.org/A003397 A003397]</span>||<span title="Numbers that are the sum of 8 positive 10th powers.">[https://oeis.org/A004808 A004808]</span>||<span title="Numbers that are the sum of 8 positive 11th powers.">[https://oeis.org/A004819 A004819]</span>
+
deo[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^3;
|-
+
deo[11] :=           (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^5;
| k=9 ||&#xa0;||<span title="Numbers that are the sum of 9 positive cubes.">[https://oeis.org/A003332 A003332]</span>||<span title="Numbers that are the sum of 9 positive 4th powers.">[https://oeis.org/A003343 A003343]</span>||<span title="Numbers that are the sum of 9 positive 5th powers.">[https://oeis.org/A003354 A003354]</span>||<span title="Numbers that are the sum of 9 positive 6th powers.">[https://oeis.org/A003365 A003365]</span>||<span title="Numbers that are the sum of 9 positive 7th powers.">[https://oeis.org/A003376 A003376]</span>||<span title="Numbers that are the sum of 9 nonzero 8th powers.">[https://oeis.org/A003387 A003387]</span>||<span title="Numbers that are the sum of 9 positive 9th powers.">[https://oeis.org/A003398 A003398]</span>||<span title="Numbers that are the sum of 9 positive 10th powers.">[https://oeis.org/A004809 A004809]</span>||<span title="Numbers that are the sum of 9 positive 11th powers.">[https://oeis.org/A004820 A004820]</span>
+
deo[12] := (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^3*(x^2-1)^3*(x-1)^4;
|-
+
Some pattern can be seen, but the higher exponents are not systematic. The problem is that some factors may have been cancelled out in the o.g.f.s polynomial fraction.  
| k=10 ||&#xa0;||<span title="Numbers that are the sum of 10 positive cubes.">[https://oeis.org/A003333 A003333]</span>||<span title="Numbers that are the sum of 10 positive 4th powers.">[https://oeis.org/A003344 A003344]</span>||<span title="Numbers that are the sum of 10 positive 5th powers.">[https://oeis.org/A003355 A003355]</span>||<span title="Numbers that are the sum of 10 positive 6th powers.">[https://oeis.org/A003366 A003366]</span>||<span title="Numbers that are the sum of 10 positive 7th powers.">[https://oeis.org/A003377 A003377]</span>||<span title="Sum of 10 nonzero 8th powers.">[https://oeis.org/A003388 A003388]</span>||<span title="Sum of 10 positive 9th powers.">[https://oeis.org/A003399 A003399]</span>||<span title="Numbers that are the sum of 10 positive 10th powers.">[https://oeis.org/A004810 A004810]</span>||<span title="Numbers that are the sum of 10 positive 11th powers.">[https://oeis.org/A004821 A004821]</span>
+
However, by multiplying both the nominator and the denominator polynomials with some factor, the following form of the denominators can be obtained:
|-
+
den[2 ] :=                                                                                             (x^2-1)  *(x-1)^2 ;
| k=11 ||&#xa0;||<span title="Numbers that are the sum of 11 positive cubes.">[https://oeis.org/A003334 A003334]</span>||<span title="Numbers that are the sum of 11 positive 4th powers.">[https://oeis.org/A003345 A003345]</span>||<span title="Numbers that are the sum of 11 positive 5th powers.">[https://oeis.org/A003356 A003356]</span>||<span title="Numbers that are the sum of 11 positive 6th powers.">[https://oeis.org/A003367 A003367]</span>||<span title="Numbers that are the sum of 11 positive 7th powers.">[https://oeis.org/A003378 A003378]</span>||<span title="Numbers that are the sum of 11 positive 8th powers.">[https://oeis.org/A003389 A003389]</span>||<span title="Sum of 11 positive 9th powers.">[https://oeis.org/A004800 A004800]</span>||<span title="Numbers that are the sum of 11 positive 10th powers.">[https://oeis.org/A004811 A004811]</span>||<span title="Numbers that are the sum of 11 positive 11th powers.">[https://oeis.org/A004822 A004822]</span>
+
den[3 ] :=                                                                                   (x^3-1)  *(x^2-1)  *(x-1)^3 ;
|-
+
den[4 ] :=                                                                        (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^4 ;
| k=12 ||&#xa0;||<span title="Numbers that are the sum of 12 positive cubes.">[https://oeis.org/A003335 A003335]</span>||<span title="Numbers that are the sum of 12 positive 4th powers.">[https://oeis.org/A003346 A003346]</span>||<span title="Numbers that are the sum of 12 positive 5th powers.">[https://oeis.org/A003357 A003357]</span>||<span title="Numbers that are the sum of 12 positive 6th powers.">[https://oeis.org/A003368 A003368]</span>||<span title="Numbers that are the sum of 12 positive 7th powers.">[https://oeis.org/A003379 A003379]</span>||<span title="Sum of 12 nonzero 8th powers.">[https://oeis.org/A003390 A003390]</span>||<span title="Sum of 12 positive 9th powers.">[https://oeis.org/A004801 A004801]</span>||<span title="Numbers that are the sum of 12 positive 10th powers.">[https://oeis.org/A004812 A004812]</span>||<span title="Numbers that are the sum of 12 positive 11th powers.">[https://oeis.org/A004823 A004823]</span>
+
den[5 ] :=                                                              (x^5-1)  *(x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^5 ;
|-
+
den[6 ] :=                                                     (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^6 ;
| k=13 ||&#xa0;||&#xa0;||<span title="Sum of 13 nonzero fourth powers.">[https://oeis.org/A047724 A047724]</span>||<span title="Sum of 13 positive 5th powers.">[https://oeis.org/A123294 A123294]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
den[7 ] :=                                             (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^7 ;
|-
+
den[8 ] :=                                    (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^8 ;
| k=14 ||&#xa0;||&#xa0;||<span title="Sum of 14 nonzero fourth powers.">[https://oeis.org/A047725 A047725]</span>||<span title="Sum of 14 positive 5th powers.">[https://oeis.org/A123295 A123295]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
den[9 ] :=                            (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^9 ;
|-
+
den[10] :=                   (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^5*(x-1)^10;
|}
+
den[11] :=           (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^5*(x-1)^11;
==== Sums of at most k positive m-th powers &gt; 0 (Table C)====
+
den[12] := (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^4*(x^2-1)^6*(x-1)^12;
{| class="wikitable" style="text-align:left"
+
Here, the exponents increase in groups of 1 for <code>(x-1)</code>, in groups of 2 for <code>(x^2-1)</code>, in groups of 3 for <code>(x^3-1)</code>, etc.
! &#xa0; !!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
+
The triangle of the exponents turns out to be A010766 (with rows reversed). The latter has a very simple formula <code>T(n,k) = floor(n/k)</code> and starts:
|-
+
  n/k  1  2  3  4  5  6  7  8  9 10 11 12
| k&lt;=2 ||&#xa0;||<span title="Numbers that are the sum of at most 2 nonzero 4th powers.">[https://oeis.org/A004831 A004831]</span>||<span title="Numbers that are the sum of at most 2 positive 5th powers.">[https://oeis.org/A004842 A004842]</span>||<span title="Numbers that are the sum of at most 2 nonzero 6th powers.">[https://oeis.org/A004853 A004853]</span>||<span title="Numbers that are the sum of at most 2 positive 7th powers.">[https://oeis.org/A004864 A004864]</span>||<span title="Numbers that are the sum of at most 2 nonzero 8th powers.">[https://oeis.org/A004875 A004875]</span>||<span title="Numbers that are the sum of at most 2 positive 9th powers.">[https://oeis.org/A004886 A004886]</span>||<span title="Numbers that are the sum of at most 2 nonzero 10th powers.">[https://oeis.org/A004897 A004897]</span>||<span title="Numbers that are the sum of at most 2 positive 11th powers.">[https://oeis.org/A004908 A004908]</span>
+
  ---+------------------------------------
|-
+
    1 | 1
| k&lt;=3 ||<span title="Numbers that are the sum of at most 3 positive cubes.">[https://oeis.org/A004825 A004825]</span>||<span title="Numbers that are the sum of at most 3 nonzero 4th powers.">[https://oeis.org/A004832 A004832]</span>||<span title="Numbers that are the sum of at most 3 positive 5th powers.">[https://oeis.org/A004843 A004843]</span>||<span title="Numbers that are the sum of at most 3 nonzero 6th powers.">[https://oeis.org/A004854 A004854]</span>||<span title="Numbers that are the sum of at most 3 positive 7th powers.">[https://oeis.org/A004865 A004865]</span>||<span title="Numbers that are the sum of at most 3 nonzero 8th powers.">[https://oeis.org/A004876 A004876]</span>||<span title="Numbers that are the sum of at most 3 positive 9th powers.">[https://oeis.org/A004887 A004887]</span>||<span title="Numbers that are the sum of at most 3 nonzero 10th powers.">[https://oeis.org/A004898 A004898]</span>||<span title="Numbers that are the sum of at most 3 positive 11th powers.">[https://oeis.org/A004909 A004909]</span>
+
    2 | 2  1
|-
+
    3 |  3  1  1
| k&lt;=4 ||<span title="Numbers that are the sum of at most 4 positive cubes.">[https://oeis.org/A004826 A004826]</span>||<span title="Numbers that are the sum of at most 4 nonzero 4th powers.">[https://oeis.org/A004833 A004833]</span>||<span title="Numbers that are the sum of at most 4 positive 5th powers.">[https://oeis.org/A004844 A004844]</span>||<span title="Numbers that are the sum of at most 4 nonzero 6th powers.">[https://oeis.org/A004855 A004855]</span>||<span title="Numbers that are the sum of at most 4 positive 7th powers.">[https://oeis.org/A004866 A004866]</span>||<span title="Numbers that are the sum of at most 4 nonzero 8th powers.">[https://oeis.org/A004877 A004877]</span>||<span title="Numbers that are the sum of at most 4 positive 9th powers.">[https://oeis.org/A004888 A004888]</span>||<span title="Numbers that are the sum of at most 4 nonzero 10th powers.">[https://oeis.org/A004899 A004899]</span>||<span title="Numbers that are the sum of at most 4 positive 11th powers.">[https://oeis.org/A004910 A004910]</span>
+
    4 |  4  2  1  1
|-
+
    5 | 5 2  1  1  1
| k&lt;=5 ||<span title="Numbers that are the sum of at most 5 positive cubes.">[https://oeis.org/A004827 A004827]</span>||<span title="Numbers that are the sum of at most 5 nonzero 4th powers.">[https://oeis.org/A004834 A004834]</span>||<span title="Numbers that are the sum of at most 5 positive 5th powers.">[https://oeis.org/A004845 A004845]</span>||<span title="Numbers that are the sum of at most 5 nonzero 6th powers.">[https://oeis.org/A004856 A004856]</span>||<span title="Numbers that are the sum of at most 5 positive 7th powers.">[https://oeis.org/A004867 A004867]</span>||<span title="Numbers that are the sum of at most 5 nonzero 8th powers.">[https://oeis.org/A004878 A004878]</span>||<span title="Numbers that are the sum of at most 5 positive 9th powers.">[https://oeis.org/A004889 A004889]</span>||<span title="Numbers that are the sum of at most 5 nonzero 10th powers.">[https://oeis.org/A004900 A004900]</span>||<span title="Numbers that are the sum of at most 5 positive 11th powers.">[https://oeis.org/A004911 A004911]</span>
+
    6 | 6  3  2  1  1  1
|-
+
    7 |  7  3  2  1  1  1  1
| k&lt;=6 ||<span title="Numbers that are the sum of at most 6 positive cubes.">[https://oeis.org/A004828 A004828]</span>||<span title="Numbers that are the sum of at most 6 nonzero 4th powers.">[https://oeis.org/A004835 A004835]</span>||<span title="Numbers that are the sum of at most 6 positive 5th powers.">[https://oeis.org/A004846 A004846]</span>||<span title="Numbers that are the sum of at most 6 nonzero 6th powers.">[https://oeis.org/A004857 A004857]</span>||<span title="Numbers that are the sum of at most 6 positive 7th powers.">[https://oeis.org/A004868 A004868]</span>||<span title="Numbers that are the sum of at most 6 nonzero 8th powers.">[https://oeis.org/A004879 A004879]</span>||<span title="Numbers that are the sum of at most 6 positive 9th powers.">[https://oeis.org/A004890 A004890]</span>||<span title="Numbers that are the sum of at most 6 nonzero 10th powers.">[https://oeis.org/A004901 A004901]</span>||<span title="Numbers that are the sum of at most 6 positive 11th powers.">[https://oeis.org/A004912 A004912]</span>
+
    8 | 8  4  2  2  1  1  1  1
|-
+
    9 | 9  4  3  2  1  1  1  1  1
| k&lt;=7 ||<span title="Numbers that are the sum of at most 7 positive cubes.">[https://oeis.org/A004829 A004829]</span>||<span title="Numbers that are the sum of at most 7 nonzero 4th powers.">[https://oeis.org/A004836 A004836]</span>||<span title="Numbers that are the sum of at most 7 positive 5th powers.">[https://oeis.org/A004847 A004847]</span>||<span title="Numbers that are the sum of at most 7 nonzero 6th powers.">[https://oeis.org/A004858 A004858]</span>||<span title="Numbers that are the sum of at most 7 positive 7th powers.">[https://oeis.org/A004869 A004869]</span>||<span title="Numbers that are the sum of at most 7 nonzero 8th powers.">[https://oeis.org/A004880 A004880]</span>||<span title="Numbers that are the sum of at most 7 positive 9th powers.">[https://oeis.org/A004891 A004891]</span>||<span title="Numbers that are the sum of at most 7 nonzero 10th powers.">[https://oeis.org/A004902 A004902]</span>||<span title="Numbers that are the sum of at most 7 positive 11th powers.">[https://oeis.org/A004913 A004913]</span>
+
  10 | 10  5  3  2  2  1  1  1  1  1
|-
+
  11 | 11  5 3  2  2  1  1  1  1  1  1
| k&lt;=8 ||<span title="Numbers that are the sum of at most 8 positive cubes.">[https://oeis.org/A004830 A004830]</span>||<span title="Numbers that are the sum of at most 8 nonzero 4th powers.">[https://oeis.org/A004837 A004837]</span>||<span title="Numbers that are the sum of at most 8 positive 5th powers.">[https://oeis.org/A004848 A004848]</span>||<span title="Numbers that are the sum of at most 8 nonzero 6th powers.">[https://oeis.org/A004859 A004859]</span>||<span title="Numbers that are the sum of at most 8 positive 7th powers.">[https://oeis.org/A004870 A004870]</span>||<span title="Numbers that are the sum of at most 8 nonzero 8th powers.">[https://oeis.org/A004881 A004881]</span>||<span title="Numbers that are the sum of at most 8 positive 9th powers.">[https://oeis.org/A004892 A004892]</span>||<span title="Numbers that are the sum of at most 8 nonzero 10th powers.">[https://oeis.org/A004903 A004903]</span>||<span title="Numbers that are the sum of at most 8 positive 11th powers.">[https://oeis.org/A004914 A004914]</span>
+
  12 | 12  6  4  3  2  2  1  1  1  1  1  1
|-
 
| k&lt;=9 ||&#xa0;||<span title="Numbers that are the sum of at most 9 nonzero 4th powers.">[https://oeis.org/A004838 A004838]</span>||<span title="Numbers that are the sum of at most 9 positive 5th powers.">[https://oeis.org/A004849 A004849]</span>||<span title="Numbers that are the sum of at most 9 nonzero 6th powers.">[https://oeis.org/A004860 A004860]</span>||<span title="Numbers that are the sum of at most 9 positive 7th powers.">[https://oeis.org/A004871 A004871]</span>||<span title="Numbers that are the sum of at most 9 nonzero 8th powers.">[https://oeis.org/A004882 A004882]</span>||<span title="Numbers that are the sum of at most 9 positive 9th powers.">[https://oeis.org/A004893 A004893]</span>||<span title="Numbers that are the sum of at most 9 nonzero 10th powers.">[https://oeis.org/A004904 A004904]</span>||<span title="Numbers that are the sum of at most 9 positive 11th powers.">[https://oeis.org/A004915 A004915]</span>
 
|-
 
| k&lt;=10 ||&#xa0;||<span title="Numbers that are the sum of at most 10 nonzero 4th powers.">[https://oeis.org/A004839 A004839]</span>||<span title="Numbers that are the sum of at most 10 positive 5th powers.">[https://oeis.org/A004850 A004850]</span>||<span title="Numbers that are the sum of at most 10 nonzero 6th powers.">[https://oeis.org/A004861 A004861]</span>||<span title="Numbers that are the sum of at most 10 positive 7th powers.">[https://oeis.org/A004872 A004872]</span>||<span title="Numbers that are the sum of at most 10 nonzero 8th powers.">[https://oeis.org/A004883 A004883]</span>||<span title="Numbers that are the sum of at most 10 positive 9th powers.">[https://oeis.org/A004894 A004894]</span>||<span title="Numbers that are the sum of at most 10 nonzero 10th powers.">[https://oeis.org/A004905 A004905]</span>||<span title="Numbers that are the sum of at most 10 positive 11th powers.">[https://oeis.org/A004916 A004916]</span>
 
|-
 
| k&lt;=11 ||&#xa0;||<span title="Numbers that are the sum of at most 11 nonzero 4th powers.">[https://oeis.org/A004840 A004840]</span>||<span title="Numbers that are the sum of at most 11 positive 5th powers.">[https://oeis.org/A004851 A004851]</span>||<span title="Numbers that are the sum of at most 11 nonzero 6th powers.">[https://oeis.org/A004862 A004862]</span>||<span title="Numbers that are the sum of at most 11 positive 7th powers.">[https://oeis.org/A004873 A004873]</span>||<span title="Numbers that are the sum of at most 11 nonzero 8th powers.">[https://oeis.org/A004884 A004884]</span>||<span title="Numbers that are the sum of at most 11 positive 9th powers.">[https://oeis.org/A004895 A004895]</span>||<span title="Numbers that are the sum of at most 11 nonzero 10th powers.">[https://oeis.org/A004906 A004906]</span>||<span title="Numbers that are the sum of at most 11 positive 11th powers.">[https://oeis.org/A004917 A004917]</span>
 
|-
 
| k&lt;=12 ||&#xa0;||<span title="Numbers that are the sum of at most 12 nonzero 4th powers.">[https://oeis.org/A004841 A004841]</span>||<span title="Numbers that are the sum of at most 12 positive 5th powers.">[https://oeis.org/A004852 A004852]</span>||<span title="Numbers that are the sum of at most 12 nonzero 6th powers.">[https://oeis.org/A004863 A004863]</span>||<span title="Numbers that are the sum of at most 12 positive 7th powers.">[https://oeis.org/A004874 A004874]</span>||<span title="Numbers that are the sum of at most 12 nonzero 8th powers.">[https://oeis.org/A004885 A004885]</span>||<span title="Numbers that are the sum of at most 12 positive 9th powers.">[https://oeis.org/A004896 A004896]</span>||<span title="Numbers that are the sum of at most 12 nonzero 10th powers.">[https://oeis.org/A004907 A004907]</span>||<span title="Numbers that are the sum of at most 12 positive 11th powers.">[https://oeis.org/A004918 A004918]</span>
 
|-
 
|}
 
==== Sums of k positive m-th powers &gt; 1 (Table D)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=2!!m=3
 
|-
 
| k=-1 ||&#xa0;||<span title="Numbers which can be written as sum of cubes &gt; 1.">[https://oeis.org/A078131 A078131]</span>
 
|-
 
| k=2 ||&#xa0;||<span title="Numbers that are the sum of 2 cubes &gt; 1.">[https://oeis.org/A294073 A294073]</span>
 
|-
 
| k=3 ||<span title="Numbers that are the sum of 3 squares &gt; 1.">[https://oeis.org/A302359 A302359]</span>||<span title="Numbers that are the sum of 3 cubes &gt; 1.">[https://oeis.org/A302360 A302360]</span>
 
|-
 
|}
 
==== Numbers that have exactly k representations as the sum of m squares &gt;= 0 (Table E)====
 
<!--A295158 quant_eq ten representations as the sum of five least_0 pow_2. -->
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=2!!&#xa0;!!&#xa0;!!m=5!!m=6!!m=7
 
|-
 
| k=1 ||&#xa0;||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly one representation as a sum of six nonnegative squares.">[https://oeis.org/A295484 A295484]</span>||&#xa0;
 
|-
 
| k=2 ||<span title="Numbers that are the sum of 2 squares in exactly 2 ways.">[https://oeis.org/A085625 A085625]</span>||&#xa0;||&#xa0;||<span title="Numbers that have exactly two representations as a sum of five nonnegative squares.">[https://oeis.org/A295150 A295150]</span>||<span title="Numbers that have exactly two representations as a sum of six nonnegative squares.">[https://oeis.org/A295485 A295485]</span>||<span title="Numbers that have exactly two representations of a sum of seven nonnegative squares.">[https://oeis.org/A295742 A295742]</span>
 
|-
 
| k=3 ||<span title="Numbers that are the sum of 2 squares in exactly 3 ways.">[https://oeis.org/A000443 A000443]</span>||&#xa0;||&#xa0;||<span title="Numbers that have exactly three representations as a sum of five nonnegative squares.">[https://oeis.org/A295151 A295151]</span>||<span title="Numbers that have exactly three representations as a sum of six nonnegative squares.">[https://oeis.org/A295486 A295486]</span>||<span title="Numbers that have exactly three representations of a sum of seven nonnegative squares.">[https://oeis.org/A295743 A295743]</span>
 
|-
 
| k=4 ||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly four representations as a sum of five nonnegative squares.">[https://oeis.org/A295152 A295152]</span>||<span title="Numbers that have exactly four representations as a sum of six nonnegative squares.">[https://oeis.org/A295487 A295487]</span>||<span title="Numbers that have exactly four representations of a sum of seven nonnegative squares.">[https://oeis.org/A295744 A295744]</span>
 
|-
 
| k=5 ||<span title="Numbers that are the sum of 2 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</span>||&#xa0;||&#xa0;||<span title="Numbers that have exactly five representations as a sum of five nonnegative squares.">[https://oeis.org/A295153 A295153]</span>||<span title="Numbers that have exactly five representations as a sum of six nonnegative squares.">[https://oeis.org/A295488 A295488]</span>||<span title="Numbers that have exactly five representations of a sum of seven nonnegative squares.">[https://oeis.org/A295745 A295745]</span>
 
|-
 
| k=6 ||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly six representations as a sum of five nonnegative squares.">[https://oeis.org/A295154 A295154]</span>||<span title="Numbers that have exactly six representations as a sum of six nonnegative squares.">[https://oeis.org/A295489 A295489]</span>||<span title="Numbers that have exactly six representations of a sum of seven nonnegative squares.">[https://oeis.org/A295747 A295747]</span>
 
|-
 
| k=7 ||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly seven representations as a sum of five nonnegative squares.">[https://oeis.org/A295155 A295155]</span>||<span title="Numbers that have exactly seven representations as a sum of six nonnegative squares.">[https://oeis.org/A295490 A295490]</span>||<span title="Numbers that have exactly seven representations of a sum of seven nonnegative squares.">[https://oeis.org/A295748 A295748]</span>
 
|-
 
| k=8 ||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly eight representations as a sum of five nonnegative squares.">[https://oeis.org/A295156 A295156]</span>||<span title="Numbers that have exactly eight representations as a sum of six nonnegative squares.">[https://oeis.org/A295491 A295491]</span>||<span title="Numbers that have exactly eight representations of a sum of seven nonnegative squares.">[https://oeis.org/A295749 A295749]</span>
 
|-
 
| k=9 ||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly nine representations as a sum of five nonnegative squares.">[https://oeis.org/A295157 A295157]</span>||<span title="Numbers that have exactly nine representations as a sum of six nonnegative squares.">[https://oeis.org/A295492 A295492]</span>||<span title="Numbers that have exactly nine representations of a sum of seven nonnegative squares.">[https://oeis.org/A295750 A295750]</span>
 
|-
 
| k=10 ||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers that have exactly ten representations as a sum of five nonnegative squares.">[https://oeis.org/A295158 A295158]</span>||<span title="Numbers that have exactly ten representations as a sum of six nonnegative squares.">[https://oeis.org/A295493 A295493]</span>||<span title="Numbers that have exactly ten representations as a sum of seven nonnegative squares.">[https://oeis.org/A295751 A295751]</span>
 
|-
 
|}
 
==New Lists==
 
===Squares===
 
==== Numbers that can be expressed as the sum of k distinct squares in m or more ways (Table R2d)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3!!m&gt;=4!!m&gt;=5!!m&gt;=6!!m&gt;=7!!m&gt;=8!!m&gt;=9!!m&gt;=10
 
|-
 
| k=1 ||<span title="The squares: a(n) = n^2.">[https://oeis.org/A000290 A000290]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=2 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of 2 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025313 A025313]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025314 A025314]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025315 A025315]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025316 A025316]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025317 A025317]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025318 A025318]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025319 A025319]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025320 A025320]</span>
 
|-
 
| k=3 ||<span title="Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 &lt;= i &lt;= j &lt;= k.">[https://oeis.org/A024796 A024796]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 2 or more ways.">[https://oeis.org/A024804 A024804]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025349 A025349]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025350 A025350]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025351 A025351]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025352 A025352]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025353 A025353]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025354 A025354]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025355 A025355]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025356 A025356]</span>
 
|-
 
| k=4 ||&#xa0;||<span title="Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.">[https://oeis.org/A259058 A259058]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025387 A025387]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025388 A025388]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025389 A025389]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025390 A025390]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025391 A025391]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025392 A025392]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025393 A025393]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025394 A025394]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k possibly equal squares in m or more ways (Table R2e)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3!!m&gt;=4!!m&gt;=5!!m&gt;=6!!m&gt;=7!!m&gt;=8!!m&gt;=9!!m&gt;=10
 
|-
 
| k=2 ||&#xa0;||<span title="Numbers that are the sum of 2 nonzero squares in 2 or more ways.">[https://oeis.org/A007692 A007692]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 3 or more ways.">[https://oeis.org/A025294 A025294]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 4 or more ways.">[https://oeis.org/A025295 A025295]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 5 or more ways.">[https://oeis.org/A025296 A025296]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 6 or more ways.">[https://oeis.org/A025297 A025297]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 7 or more ways.">[https://oeis.org/A025298 A025298]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 8 or more ways.">[https://oeis.org/A025299 A025299]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 9 or more ways.">[https://oeis.org/A025300 A025300]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 10 or more ways.">[https://oeis.org/A025301 A025301]</span>
 
|-
 
| k=3 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of 3 nonzero squares in 3 or more ways.">[https://oeis.org/A025331 A025331]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 4 or more ways.">[https://oeis.org/A025332 A025332]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 5 or more ways.">[https://oeis.org/A025333 A025333]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 6 or more ways.">[https://oeis.org/A025334 A025334]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 7 or more ways.">[https://oeis.org/A025335 A025335]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 8 or more ways.">[https://oeis.org/A025336 A025336]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 9 or more ways.">[https://oeis.org/A025337 A025337]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 10 or more ways.">[https://oeis.org/A025338 A025338]</span>
 
|-
 
| k=4 ||&#xa0;||<span title="Numbers that are the sum of 4 nonzero squares in 2 or more ways.">[https://oeis.org/A025367 A025367]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 3 or more ways.">[https://oeis.org/A025368 A025368]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 4 or more ways.">[https://oeis.org/A025369 A025369]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 5 or more ways.">[https://oeis.org/A025370 A025370]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 6 or more ways.">[https://oeis.org/A025371 A025371]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 7 or more ways.">[https://oeis.org/A025372 A025372]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 8 or more ways.">[https://oeis.org/A025373 A025373]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 9 or more ways.">[https://oeis.org/A025374 A025374]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 10 or more ways.">[https://oeis.org/A025375 A025375]</span>
 
|-
 
| k=5 ||&#xa0;||<span title="Numbers that are the sum of five squares in two or more ways.">[https://oeis.org/A344795 A344795]</span>||<span title="Numbers that are the sum of five squares in three or more ways.">[https://oeis.org/A344796 A344796]</span>||<span title="Numbers that are the sum of five squares in four or more ways.">[https://oeis.org/A344797 A344797]</span>||<span title="Numbers that are the sum of five squares in five or more ways.">[https://oeis.org/A344798 A344798]</span>||<span title="Numbers that are the sum of five squares in six or more ways.">[https://oeis.org/A344799 A344799]</span>||<span title="Numbers that are the sum of five squares in seven or more ways.">[https://oeis.org/A344800 A344800]</span>||<span title="Numbers that are the sum of five squares in eight or more ways.">[https://oeis.org/A344801 A344801]</span>||<span title="Numbers that are the sum of five squares in nine or more ways.">[https://oeis.org/A344802 A344802]</span>||<span title="Numbers that are the sum of five squares in ten or more ways.">[https://oeis.org/A344803 A344803]</span>
 
|-
 
| k=6 ||<span title="Numbers that are the sum of six second powers in one or more ways.">[https://oeis.org/A344805 A344805]</span>||<span title="Numbers that are the sum of six squares in two or more ways.">[https://oeis.org/A344806 A344806]</span>||<span title="Numbers that are the sum of six squares in three or more ways.">[https://oeis.org/A344807 A344807]</span>||<span title="Numbers that are the sum of six squares in four or more ways.">[https://oeis.org/A344808 A344808]</span>||<span title="Numbers that are the sum of six squares in five or more ways.">[https://oeis.org/A344809 A344809]</span>||<span title="Numbers that are the sum of six squares in six or more ways.">[https://oeis.org/A344810 A344810]</span>||<span title="Numbers that are the sum of six squares in seven or more ways.">[https://oeis.org/A344811 A344811]</span>||<span title="Numbers that are the sum of six squares in eight or more ways.">[https://oeis.org/A344812 A344812]</span>||<span title="Numbers that are the sum of six squares in nine or more ways.">[https://oeis.org/A345476 A345476]</span>||<span title="Numbers that are the sum of six squares in ten or more ways.">[https://oeis.org/A345477 A345477]</span>
 
|-
 
| k=7 ||<span title="Numbers that are the sum of seven squares in one or more ways.">[https://oeis.org/A345478 A345478]</span>||<span title="Numbers that are the sum of seven squares in two or more ways.">[https://oeis.org/A345479 A345479]</span>||<span title="Numbers that are the sum of seven squares in three or more ways.">[https://oeis.org/A345480 A345480]</span>||<span title="Numbers that are the sum of seven squares in four or more ways.">[https://oeis.org/A345481 A345481]</span>||<span title="Numbers that are the sum of seven squares in five or more ways.">[https://oeis.org/A345482 A345482]</span>||<span title="Numbers that are the sum of seven squares in six or more ways.">[https://oeis.org/A345483 A345483]</span>||<span title="Numbers that are the sum of seven squares in seven or more ways.">[https://oeis.org/A345484 A345484]</span>||<span title="Numbers that are the sum of seven squares in eight or more ways.">[https://oeis.org/A345485 A345485]</span>||<span title="Numbers that are the sum of seven squares in nine or more ways.">[https://oeis.org/A345486 A345486]</span>||<span title="Numbers that are the sum of seven squares in ten or more ways.">[https://oeis.org/A345487 A345487]</span>
 
|-
 
| k=8 ||<span title="Numbers that are the sum of eight squares in one or more ways.">[https://oeis.org/A345488 A345488]</span>||<span title="Numbers that are the sum of eight squares in two or more ways.">[https://oeis.org/A345489 A345489]</span>||<span title="Numbers that are the sum of eight squares in three or more ways.">[https://oeis.org/A345490 A345490]</span>||<span title="Numbers that are the sum of eight squares in four or more ways.">[https://oeis.org/A345491 A345491]</span>||<span title="Numbers that are the sum of eight squares in five or more ways.">[https://oeis.org/A345492 A345492]</span>||<span title="Numbers that are the sum of eight squares in six or more ways.">[https://oeis.org/A345493 A345493]</span>||<span title="Numbers that are the sum of eight squares in seven or more ways.">[https://oeis.org/A345494 A345494]</span>||<span title="Numbers that are the sum of eight squares in eight or more ways.">[https://oeis.org/A345495 A345495]</span>||<span title="Numbers that are the sum of eight squares in nine or more ways.">[https://oeis.org/A345496 A345496]</span>||<span title="Numbers that are the sum of eight squares in ten or more ways.">[https://oeis.org/A345497 A345497]</span>
 
|-
 
| k=9 ||<span title="Numbers that are the sum of nine squares in one or more ways.">[https://oeis.org/A345498 A345498]</span>||<span title="Numbers that are the sum of nine squares in two or more ways.">[https://oeis.org/A345499 A345499]</span>||<span title="Numbers that are the sum of nine squares in three or more ways.">[https://oeis.org/A345500 A345500]</span>||<span title="Numbers that are the sum of nine squares in four or more ways.">[https://oeis.org/A345501 A345501]</span>||<span title="Numbers that are the sum of nine squares in five or more ways.">[https://oeis.org/A345502 A345502]</span>||<span title="Numbers that are the sum of nine squares in six or more ways.">[https://oeis.org/A345503 A345503]</span>||<span title="Numbers that are the sum of nine squares in seven or more ways.">[https://oeis.org/A345504 A345504]</span>||<span title="Numbers that are the sum of nine squares in eight or more ways.">[https://oeis.org/A345505 A345505]</span>||&#xa0;||<span title="Numbers that are the sum of nine squares in ten or more ways.">[https://oeis.org/A346803 A346803]</span>
 
|-
 
| k=10 ||<span title="Numbers that are the sum of ten squares in one or more ways.">[https://oeis.org/A345508 A345508]</span>||<span title="Numbers that are the sum of ten squares in two or more ways.">[https://oeis.org/A345509 A345509]</span>||<span title="Numbers that are the sum of ten squares in three or more ways.">[https://oeis.org/A345510 A345510]</span>||&#xa0;||<span title="Numbers that are the sum of ten squares in five or more ways.">[https://oeis.org/A346804 A346804]</span>||<span title="Numbers that are the sum of ten squares in six or more ways.">[https://oeis.org/A346805 A346805]</span>||<span title="Numbers that are the sum of ten squares in seven or more ways.">[https://oeis.org/A346806 A346806]</span>||<span title="Numbers that are the sum of ten squares in eight or more ways.">[https://oeis.org/A346807 A346807]</span>||&#xa0;||<span title="Numbers that are the sum of ten squares in ten or more ways.">[https://oeis.org/A346808 A346808]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k distinct squares in exactly m ways (Table S2d)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
 
|-
 
| k=2 ||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025302 A025302]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025303 A025303]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025304 A025304]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025305 A025305]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025306 A025306]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025307 A025307]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025308 A025308]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025309 A025309]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025310 A025310]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025311 A025311]</span>
 
|-
 
| k=3 ||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly one way.">[https://oeis.org/A025339 A025339]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025340 A025340]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025341 A025341]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025342 A025342]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025343 A025343]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025344 A025344]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025345 A025345]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025346 A025346]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025347 A025347]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025348 A025348]</span>
 
|-
 
| k=4 ||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025376 A025376]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025377 A025377]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025378 A025378]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025379 A025379]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025380 A025380]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025381 A025381]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025382 A025382]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025383 A025383]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025384 A025384]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025385 A025385]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k possibly equal squares in exactly m ways (Table S2e)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
 
|-
 
| k=2 ||<span title="Numbers that are the sum of 2 nonzero squares in exactly 1 way.">[https://oeis.org/A025284 A025284]</span>||<span title="Numbers that are the sum of 2 squares in exactly 2 ways.">[https://oeis.org/A085625 A085625]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 3 ways.">[https://oeis.org/A025286 A025286]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 4 ways.">[https://oeis.org/A025287 A025287]</span>||<span title="Numbers that are the sum of 2 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 6 ways.">[https://oeis.org/A025289 A025289]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 7 ways.">[https://oeis.org/A025290 A025290]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 8 ways.">[https://oeis.org/A025291 A025291]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 9 ways.">[https://oeis.org/A025292 A025292]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 10 ways.">[https://oeis.org/A025293 A025293]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 11 ways.">[https://oeis.org/A236711 A236711]</span>
 
|-
 
| k=3 ||<span title="Numbers that are the sum of 3 nonzero squares in exactly 1 way.">[https://oeis.org/A025321 A025321]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 2 ways.">[https://oeis.org/A025322 A025322]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 3 ways.">[https://oeis.org/A025323 A025323]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 4 ways.">[https://oeis.org/A025324 A025324]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 5 ways.">[https://oeis.org/A025325 A025325]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 6 ways.">[https://oeis.org/A025326 A025326]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 7 ways.">[https://oeis.org/A025327 A025327]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 8 ways.">[https://oeis.org/A025328 A025328]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 9 ways.">[https://oeis.org/A025329 A025329]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 10 ways.">[https://oeis.org/A025330 A025330]</span>||&#xa0;
 
|-
 
| k=4 ||<span title="Numbers that are the sum of 4 nonzero squares in exactly 1 way.">[https://oeis.org/A025357 A025357]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 2 ways.">[https://oeis.org/A025358 A025358]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 3 ways.">[https://oeis.org/A025359 A025359]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 4 ways.">[https://oeis.org/A025360 A025360]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 5 ways.">[https://oeis.org/A025361 A025361]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 6 ways.">[https://oeis.org/A025362 A025362]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 7 ways.">[https://oeis.org/A025363 A025363]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 8 ways.">[https://oeis.org/A025364 A025364]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 9 ways.">[https://oeis.org/A025365 A025365]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 10 ways.">[https://oeis.org/A025366 A025366]</span>||&#xa0;
 
|-
 
| k=5 ||<span title="Numbers that are the sum of 5 nonzero squares in exactly 1 way.">[https://oeis.org/A294675 A294675]</span>||<span title="Numbers that have exactly two representations as a sum of five nonnegative squares.">[https://oeis.org/A295150 A295150]</span>||<span title="Numbers that have exactly three representations as a sum of five nonnegative squares.">[https://oeis.org/A295151 A295151]</span>||<span title="Numbers that have exactly four representations as a sum of five nonnegative squares.">[https://oeis.org/A295152 A295152]</span>||<span title="Numbers that have exactly five representations as a sum of five nonnegative squares.">[https://oeis.org/A295153 A295153]</span>||<span title="Numbers that have exactly six representations as a sum of five nonnegative squares.">[https://oeis.org/A295154 A295154]</span>||<span title="Numbers that have exactly seven representations as a sum of five nonnegative squares.">[https://oeis.org/A295155 A295155]</span>||<span title="Numbers that have exactly eight representations as a sum of five nonnegative squares.">[https://oeis.org/A295156 A295156]</span>||<span title="Numbers that have exactly nine representations as a sum of five nonnegative squares.">[https://oeis.org/A295157 A295157]</span>||<span title="Numbers that have exactly ten representations as a sum of five nonnegative squares.">[https://oeis.org/A295158 A295158]</span>||&#xa0;
 
|-
 
| k=6 ||<span title="Numbers that have exactly one representation as a sum of six positive squares.">[https://oeis.org/A295670 A295670]</span>||<span title="Numbers that have exactly two representations as a sum of six positive squares.">[https://oeis.org/A295692 A295692]</span>||<span title="Numbers that have exactly three representations as a sum of six positive squares.">[https://oeis.org/A295693 A295693]</span>||<span title="Numbers that have exactly four representations as a sum of six positive squares.">[https://oeis.org/A295694 A295694]</span>||<span title="Numbers that have exactly five representations as a sum of six positive squares.">[https://oeis.org/A295695 A295695]</span>||<span title="Numbers that have exactly six representations as a sum of six positive squares.">[https://oeis.org/A295696 A295696]</span>||<span title="Numbers that have exactly seven representations as a sum of six positive squares.">[https://oeis.org/A295697 A295697]</span>||<span title="Numbers that have exactly eight representations as a sum of six positive squares.">[https://oeis.org/A295698 A295698]</span>||<span title="Numbers that have exactly nine representations as a sum of six positive squares.">[https://oeis.org/A295699 A295699]</span>||<span title="Numbers that have exactly ten representations as a sum of six positive squares.">[https://oeis.org/A295700 A295700]</span>||&#xa0;
 
|-
 
| k=7 ||<span title="Numbers that have exactly one representation as a sum of seven positive squares.">[https://oeis.org/A295797 A295797]</span>||<span title="Numbers that have exactly two representations as a sum of seven positive squares.">[https://oeis.org/A295799 A295799]</span>||<span title="Numbers that have exactly three representations as a sum of seven positive squares.">[https://oeis.org/A295800 A295800]</span>||<span title="Numbers that have exactly four representations as a sum of seven positive squares.">[https://oeis.org/A295801 A295801]</span>||<span title="Numbers that have exactly five representations as a sum of seven positive squares.">[https://oeis.org/A295802 A295802]</span>||<span title="Numbers that have exactly six representations as a sum of seven positive squares.">[https://oeis.org/A295803 A295803]</span>||<span title="Numbers that have exactly seven representations as a sum of seven positive squares.">[https://oeis.org/A295804 A295804]</span>||<span title="Numbers that have exactly eight representations as a sum of seven positive squares.">[https://oeis.org/A295805 A295805]</span>||<span title="Numbers that have exactly nine representations as a sum of seven positive squares.">[https://oeis.org/A295806 A295806]</span>||<span title="Numbers that have exactly ten representations as a sum of seven positive squares.">[https://oeis.org/A295807 A295807]</span>||&#xa0;
 
|-
 
|}
 
  
=== Cubes ===
+
In order to support this hypothesis, the following steps were performed:
==== Numbers that can be expressed as the sum of k distinct cubes in m or more ways (Table R3d)====
+
# Determine the denomiator polynomials <code>den</code> above, and take their coefficients as the signature for a linear recurrence with constant coefficients.
{| class="wikitable" style="text-align:left"
+
# Compute a sufficient number of terms (150 - 270) for the column sequences A055278-A055287 with Michale Somos' PARI program in A055277. This took about 1 day.
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3
+
# Take <code>degree(den[n])</code> initial terms from step 2, run the recurrence and verify it for all terms.
|-
 
| k=1 ||<span title="The cubes: a(n) = n^3.">[https://oeis.org/A000578 A000578]</span>||&#xa0;||&#xa0;
 
|-
 
| k=2 ||<span title="Taxi-cab numbers: sums of 2 cubes in more than 1 way.">[https://oeis.org/A001235 A001235]</span>||&#xa0;||&#xa0;
 
|-
 
| k=3 ||&#xa0;||<span title="Numbers that are the sum of 3 distinct positive cubes in 2 or more ways.">[https://oeis.org/A024974 A024974]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in 3 or more ways.">[https://oeis.org/A025402 A025402]</span>
 
|-
 
| k=4 ||&#xa0;||<span title="Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes.">[https://oeis.org/A259060 A259060]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in 3 or more ways.">[https://oeis.org/A025413 A025413]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k possibly equal cubes in m or more ways (Table R3e)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3!!m&gt;=4!!m&gt;=5!!m&gt;=6!!m&gt;=7!!m&gt;=8!!m&gt;=9!!m&gt;=10
 
|-
 
| k=2 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of two positive cubes in at least three ways (all solutions).">[https://oeis.org/A018787 A018787]</span>||<span title="Numbers that are the sum of two positive cubes in at least four ways (all solutions).">[https://oeis.org/A023051 A023051]</span>||<span title="Sum of two positive cubes in at least five ways (all solutions).">[https://oeis.org/A051167 A051167]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=3 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of 3 positive cubes in 3 or more ways.">[https://oeis.org/A025398 A025398]</span>||<span title="Numbers that are the sum of three positive cubes in four or more ways.">[https://oeis.org/A343968 A343968]</span>||<span title="Numbers that are the sum of three positive cubes in five or more ways.">[https://oeis.org/A343967 A343967]</span>||<span title="Numbers that are the sum of three third powers in six or more ways.">[https://oeis.org/A345083 A345083]</span>||<span title="Numbers that are the sum of three third powers in seven or more ways.">[https://oeis.org/A345086 A345086]</span>||<span title="Numbers that are the sum of three third powers in eight or more ways.">[https://oeis.org/A345087 A345087]</span>||<span title="Numbers that are the sum of three third powers in nine or more ways.">[https://oeis.org/A345119 A345119]</span>||<span title="Numbers that are the sum of three third powers in ten or more ways.">[https://oeis.org/A345121 A345121]</span>
 
|-
 
| k=4 ||<span title="Numbers that are the sum of 4 positive cubes in 1 or more way.">[https://oeis.org/A003327 A003327]</span>||<span title="Numbers that are the sum of 4 positive cubes in 2 or more ways.">[https://oeis.org/A025406 A025406]</span>||<span title="Numbers that are the sum of 4 positive cubes in 3 or more ways.">[https://oeis.org/A025407 A025407]</span>||<span title="Numbers that are the sum of four positive cubes in four or more ways.">[https://oeis.org/A343971 A343971]</span>||<span title="Numbers that are the sum of four positive cubes in five or more ways.">[https://oeis.org/A343987 A343987]</span>||<span title="Numbers that are the sum of four third powers in six or more ways.">[https://oeis.org/A345148 A345148]</span>||<span title="Numbers that are the sum of four third powers in seven or more ways.">[https://oeis.org/A345150 A345150]</span>||<span title="Numbers that are the sum of four third powers in eight or more ways.">[https://oeis.org/A345152 A345152]</span>||<span title="Numbers that are the sum of four third powers in nine or more ways.">[https://oeis.org/A345146 A345146]</span>||<span title="Numbers that are the sum of four third powers in ten or more ways.">[https://oeis.org/A345155 A345155]</span>
 
|-
 
| k=5 ||&#xa0;||<span title="Numbers that are the sum of five positive cubes in two or more ways.">[https://oeis.org/A343702 A343702]</span>||<span title="Numbers that are the sum of five positive cubes in three or more ways.">[https://oeis.org/A343704 A343704]</span>||<span title="Numbers that are the sum of five positive cubes in four or more ways.">[https://oeis.org/A344034 A344034]</span>||<span title="Numbers that are the sum of five positive cubes in five or more ways.">[https://oeis.org/A343989 A343989]</span>||<span title="Numbers that are the sum of five third powers in six or more ways.">[https://oeis.org/A345174 A345174]</span>||<span title="Numbers that are the sum of five third powers in seven or more ways.">[https://oeis.org/A345180 A345180]</span>||<span title="Numbers that are the sum of five third powers in eight or more ways.">[https://oeis.org/A345183 A345183]</span>||<span title="Numbers that are the sum of five third powers in nine or more ways.">[https://oeis.org/A345185 A345185]</span>||<span title="Numbers that are the sum of five third powers in ten or more ways.">[https://oeis.org/A345187 A345187]</span>
 
|-
 
| k=6 ||&#xa0;||<span title="Numbers that are the sum of six cubes in two or more ways.">[https://oeis.org/A345511 A345511]</span>||<span title="Numbers that are the sum of six cubes in three or more ways.">[https://oeis.org/A345512 A345512]</span>||<span title="Numbers that are the sum of six cubes in four or more ways.">[https://oeis.org/A345513 A345513]</span>||<span title="Numbers that are the sum of six cubes in five or more ways.">[https://oeis.org/A345514 A345514]</span>||<span title="Numbers that are the sum of six cubes in six or more ways.">[https://oeis.org/A345515 A345515]</span>||<span title="Numbers that are the sum of six cubes in seven or more ways.">[https://oeis.org/A345516 A345516]</span>||<span title="Numbers that are the sum of six cubes in eight or more ways.">[https://oeis.org/A345517 A345517]</span>||<span title="Numbers that are the sum of six cubes in nine or more ways.">[https://oeis.org/A345518 A345518]</span>||<span title="Numbers that are the sum of six cubes in ten or more ways.">[https://oeis.org/A345519 A345519]</span>
 
|-
 
| k=7 ||&#xa0;||<span title="Numbers that are the sum of seven cubes in two or more ways.">[https://oeis.org/A345520 A345520]</span>||<span title="Numbers that are the sum of seven cubes in three or more ways.">[https://oeis.org/A345521 A345521]</span>||<span title="Numbers that are the sum of seven cubes in four or more ways.">[https://oeis.org/A345522 A345522]</span>||<span title="Numbers that are the sum of seven cubes in five or more ways.">[https://oeis.org/A345523 A345523]</span>||<span title="Numbers that are the sum of seven cubes in six or more ways.">[https://oeis.org/A345524 A345524]</span>||<span title="Numbers that are the sum of seven cubes in seven or more ways.">[https://oeis.org/A345525 A345525]</span>||<span title="Numbers that are the sum of seven cubes in eight or more ways.">[https://oeis.org/A345526 A345526]</span>||<span title="Numbers that are the sum of seven cubes in nine or more ways.">[https://oeis.org/A345527 A345527]</span>||<span title="Numbers that are the sum of seven cubes in ten or more ways.">[https://oeis.org/A345506 A345506]</span>
 
|-
 
| k=8 ||&#xa0;||<span title="Numbers that are the sum of eight cubes in two or more ways.">[https://oeis.org/A345532 A345532]</span>||<span title="Numbers that are the sum of eight cubes in three or more ways.">[https://oeis.org/A345533 A345533]</span>||<span title="Numbers that are the sum of eight cubes in four or more ways.">[https://oeis.org/A345534 A345534]</span>||<span title="Numbers that are the sum of eight cubes in five or more ways.">[https://oeis.org/A345535 A345535]</span>||<span title="Numbers that are the sum of eight cubes in six or more ways.">[https://oeis.org/A345536 A345536]</span>||<span title="Numbers that are the sum of eight cubes in seven or more ways.">[https://oeis.org/A345537 A345537]</span>||<span title="Numbers that are the sum of eight cubes in eight or more ways.">[https://oeis.org/A345538 A345538]</span>||<span title="Numbers that are the sum of eight cubes in nine or more ways.">[https://oeis.org/A345539 A345539]</span>||<span title="Numbers that are the sum of eight cubes in ten or more ways.">[https://oeis.org/A345540 A345540]</span>
 
|-
 
| k=9 ||&#xa0;||<span title="Numbers that are the sum of nine cubes in two or more ways.">[https://oeis.org/A345541 A345541]</span>||<span title="Numbers that are the sum of nine cubes in three or more ways.">[https://oeis.org/A345542 A345542]</span>||<span title="Numbers that are the sum of nine cubes in four or more ways.">[https://oeis.org/A345543 A345543]</span>||<span title="Numbers that are the sum of nine cubes in five or more ways.">[https://oeis.org/A345544 A345544]</span>||<span title="Numbers that are the sum of nine cubes in six or more ways.">[https://oeis.org/A345545 A345545]</span>||<span title="Numbers that are the sum of nine cubes in seven or more ways.">[https://oeis.org/A345546 A345546]</span>||<span title="Numbers that are the sum of nine cubes in eight or more ways.">[https://oeis.org/A345547 A345547]</span>||<span title="Numbers that are the sum of nine cubes in nine or more ways.">[https://oeis.org/A345548 A345548]</span>||<span title="Numbers that are the sum of nine cubes in ten or more ways.">[https://oeis.org/A345549 A345549]</span>
 
|-
 
| k=10 ||&#xa0;||<span title="Numbers that are the sum of ten cubes in two or more ways.">[https://oeis.org/A345550 A345550]</span>||<span title="Numbers that are the sum of ten cubes in three or more ways.">[https://oeis.org/A345551 A345551]</span>||<span title="Numbers that are the sum of ten cubes in four or more ways.">[https://oeis.org/A345552 A345552]</span>||<span title="Numbers that are the sum of ten cubes in five or more ways.">[https://oeis.org/A345553 A345553]</span>||<span title="Numbers that are the sum of ten cubes in six or more ways.">[https://oeis.org/A345554 A345554]</span>||<span title="Numbers that are the sum of ten cubes in seven or more ways.">[https://oeis.org/A345555 A345555]</span>||<span title="Numbers that are the sum of ten cubes in eight or more ways.">[https://oeis.org/A345556 A345556]</span>||<span title="Numbers that are the sum of ten cubes in nine or more ways.">[https://oeis.org/A345557 A345557]</span>||<span title="Numbers that are the sum of ten cubes in ten or more ways.">[https://oeis.org/A345558 A345558]</span>
 
|-
 
|}
 
  
==== Numbers that can be expressed as the sum of k distinct cubes in exactly m ways (Table S3d)====
+
In fact, the initial terms could be prefixed by <code>n</code> zeros.
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=1!!m=2!!m=3
 
|-
 
| k=3 ||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.">[https://oeis.org/A025399 A025399]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 2 ways.">[https://oeis.org/A025400 A025400]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 3 ways.">[https://oeis.org/A025401 A025401]</span>
 
|-
 
| k=4 ||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 1 way.">[https://oeis.org/A025408 A025408]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 2 ways.">[https://oeis.org/A025409 A025409]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 3 ways.">[https://oeis.org/A025410 A025410]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k possibly equal cubes in exactly m ways (Table S3e)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
 
|-
 
| k=2 ||<span title="Numbers that are the sum of two positive cubes in exactly one way.">[https://oeis.org/A338667 A338667]</span>||<span title="Numbers that are the sum of two positive cubes in exactly two ways.">[https://oeis.org/A343708 A343708]</span>||<span title="Numbers that are the sum of two cubes in exactly three ways.">[https://oeis.org/A344804 A344804]</span>||<span title="Numbers that are the sum of two cubes in exactly four ways.">[https://oeis.org/A345865 A345865]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=3 ||<span title="Numbers that are the sum of 3 positive cubes in exactly 1 way.">[https://oeis.org/A025395 A025395]</span>||<span title="Numbers that are the sum of 3 positive cubes in exactly 2 ways.">[https://oeis.org/A025396 A025396]</span>||<span title="Numbers that are the sum of 3 positive cubes in exactly 3 ways.">[https://oeis.org/A025397 A025397]</span>||<span title="Numbers that are the sum of three positive cubes in exactly 4 ways.">[https://oeis.org/A343969 A343969]</span>||<span title="Numbers that are the sum of three positive cubes in exactly five ways.">[https://oeis.org/A343970 A343970]</span>||<span title="Numbers that are the sum of three third powers in exactly six ways.">[https://oeis.org/A345084 A345084]</span>||<span title="Numbers that are the sum of three third powers in exactly seven ways.">[https://oeis.org/A345085 A345085]</span>||<span title="Numbers that are the sum of three third powers in exactly eight ways.">[https://oeis.org/A345088 A345088]</span>||<span title="Numbers that are the sum of three third powers in exactly nine ways.">[https://oeis.org/A345120 A345120]</span>||<span title="Numbers that are the sum of three third powers in exactly ten ways.">[https://oeis.org/A345122 A345122]</span>
 
|-
 
| k=4 ||<span title="Numbers that are the sum of 4 positive cubes in exactly 1 way.">[https://oeis.org/A025403 A025403]</span>||<span title="Numbers that are the sum of 4 positive cubes in exactly 2 ways.">[https://oeis.org/A025404 A025404]</span>||<span title="Numbers that are the sum of 4 positive cubes in exactly 3 ways.">[https://oeis.org/A025405 A025405]</span>||<span title="Numbers that are the sum of four positive cubes in exactly four ways.">[https://oeis.org/A343972 A343972]</span>||<span title="Numbers that are the sum of four positive cubes in exactly five ways.">[https://oeis.org/A343988 A343988]</span>||<span title="Numbers that are the sum of four third powers in exactly six ways.">[https://oeis.org/A345149 A345149]</span>||<span title="Numbers that are the sum of four third powers in exactly seven ways.">[https://oeis.org/A345151 A345151]</span>||<span title="Numbers that are the sum of four third powers in exactly eight ways.">[https://oeis.org/A345153 A345153]</span>||<span title="Numbers that are the sum of four third powers in exactly nine ways.">[https://oeis.org/A345154 A345154]</span>||<span title="Numbers that are the sum of four third powers in exactly ten ways.">[https://oeis.org/A345156 A345156]</span>
 
|-
 
| k=5 ||<span title="Numbers that are the sum of 5 positive cubes in exactly 1 way.">[https://oeis.org/A048926 A048926]</span>||<span title="Numbers that are the sum of 5 positive cubes in exactly 2 ways.">[https://oeis.org/A048927 A048927]</span>||<span title="Numbers that are the sum of five positive cubes in exactly three ways.">[https://oeis.org/A343705 A343705]</span>||<span title="Numbers that are the sum of five positive cubes in exactly four ways.">[https://oeis.org/A344035 A344035]</span>||&#xa0;||<span title="Numbers that are the sum of five third powers in exactly six ways.">[https://oeis.org/A345175 A345175]</span>||<span title="Numbers that are the sum of five third powers in exactly seven ways.">[https://oeis.org/A345181 A345181]</span>||<span title="Numbers that are the sum of five third powers in exactly eight ways.">[https://oeis.org/A345184 A345184]</span>||<span title="Numbers that are the sum of five third powers in exactly nine ways.">[https://oeis.org/A345186 A345186]</span>||<span title="Numbers that are the sum of five third powers in exactly ten ways.">[https://oeis.org/A345188 A345188]</span>
 
|-
 
| k=6 ||<span title="Numbers that are the sum of 6 positive cubes in exactly 1 way.">[https://oeis.org/A048929 A048929]</span>||<span title="Numbers that are the sum of 6 positive cubes in exactly 2 ways.">[https://oeis.org/A048930 A048930]</span>||<span title="Numbers that are the sum of 6 positive cubes in exactly 3 ways.">[https://oeis.org/A048931 A048931]</span>||<span title="Numbers that are the sum of six cubes in exactly four ways.">[https://oeis.org/A345766 A345766]</span>||<span title="Numbers that are the sum of six cubes in exactly five ways.">[https://oeis.org/A345767 A345767]</span>||<span title="Numbers that are the sum of six cubes in exactly six ways.">[https://oeis.org/A345768 A345768]</span>||<span title="Numbers that are the sum of six cubes in exactly seven ways.">[https://oeis.org/A345769 A345769]</span>||<span title="Numbers that are the sum of six cubes in exactly eight ways.">[https://oeis.org/A345770 A345770]</span>||<span title="Numbers that are the sum of six cubes in exactly nine ways.">[https://oeis.org/A345771 A345771]</span>||<span title="Numbers that are the sum of six cubes in exactly ten ways.">[https://oeis.org/A345772 A345772]</span>
 
|-
 
| k=7 ||<span title="Numbers that are the sum of seven cubes in exactly one way.">[https://oeis.org/A345773 A345773]</span>||<span title="Numbers that are the sum of seven cubes in exactly two ways.">[https://oeis.org/A345774 A345774]</span>||<span title="Numbers that are the sum of seven cubes in exactly three ways.">[https://oeis.org/A345775 A345775]</span>||<span title="Numbers that are the sum of seven cubes in exactly four ways.">[https://oeis.org/A345776 A345776]</span>||<span title="Numbers that are the sum of seven cubes in exactly five ways.">[https://oeis.org/A345777 A345777]</span>||<span title="Numbers that are the sum of seven cubes in exactly six ways.">[https://oeis.org/A345778 A345778]</span>||<span title="Numbers that are the sum of seven cubes in exactly seven ways.">[https://oeis.org/A345779 A345779]</span>||<span title="Numbers that are the sum of seven cubes in exactly eight ways.">[https://oeis.org/A345780 A345780]</span>||<span title="Numbers that are the sum of seven cubes in exactly nine ways.">[https://oeis.org/A345781 A345781]</span>||<span title="Numbers that are the sum of seven cubes in exactly ten ways.">[https://oeis.org/A345782 A345782]</span>
 
|-
 
| k=8 ||<span title="Numbers that are the sum of eight cubes in exactly one way.">[https://oeis.org/A345783 A345783]</span>||<span title="Numbers that are the sum of eight cubes in exactly two ways.">[https://oeis.org/A345784 A345784]</span>||<span title="Numbers that are the sum of eight cubes in exactly three ways.">[https://oeis.org/A345785 A345785]</span>||<span title="Numbers that are the sum of eight cubes in exactly four ways.">[https://oeis.org/A345786 A345786]</span>||<span title="Numbers that are the sum of eight cubes in exactly five ways.">[https://oeis.org/A345787 A345787]</span>||<span title="Numbers that are the sum of eight cubes in exactly six ways.">[https://oeis.org/A345788 A345788]</span>||<span title="Numbers that are the sum of eight cubes in exactly seven ways.">[https://oeis.org/A345789 A345789]</span>||<span title="Numbers that are the sum of eight cubes in exactly eight ways.">[https://oeis.org/A345790 A345790]</span>||<span title="Numbers that are the sum of eight cubes in exactly nine ways.">[https://oeis.org/A345791 A345791]</span>||<span title="Numbers that are the sum of eight cubes in exactly ten ways.">[https://oeis.org/A345792 A345792]</span>
 
|-
 
| k=9 ||<span title="Numbers that are the sum of nine cubes in exactly one way.">[https://oeis.org/A345793 A345793]</span>||<span title="Numbers that are the sum of nine cubes in exactly two ways.">[https://oeis.org/A345794 A345794]</span>||<span title="Numbers that are the sum of nine cubes in exactly three ways.">[https://oeis.org/A345795 A345795]</span>||<span title="Numbers that are the sum of nine cubes in exactly four ways.">[https://oeis.org/A345796 A345796]</span>||<span title="Numbers that are the sum of nine cubes in exactly five ways.">[https://oeis.org/A345797 A345797]</span>||<span title="Numbers that are the sum of nine cubes in exactly six ways.">[https://oeis.org/A345798 A345798]</span>||<span title="Numbers that are the sum of nine cubes in exactly seven ways.">[https://oeis.org/A345799 A345799]</span>||<span title="Numbers that are the sum of nine cubes in exactly eight ways.">[https://oeis.org/A345800 A345800]</span>||<span title="Numbers that are the sum of nine cubes in exactly nine ways.">[https://oeis.org/A345801 A345801]</span>||<span title="Numbers that are the sum of nine cubes in exactly ten ways.">[https://oeis.org/A345802 A345802]</span>
 
|-
 
| k=10 ||<span title="Numbers that are the sum of ten cubes in exactly one ways.">[https://oeis.org/A345803 A345803]</span>||<span title="Numbers that are the sum of ten cubes in exactly two ways.">[https://oeis.org/A345804 A345804]</span>||<span title="Numbers that are the sum of ten cubes in exactly three ways.">[https://oeis.org/A345805 A345805]</span>||<span title="Numbers that are the sum of ten cubes in exactly four ways.">[https://oeis.org/A345806 A345806]</span>||<span title="Numbers that are the sum of ten cubes in exactly five ways.">[https://oeis.org/A345807 A345807]</span>||<span title="Numbers that are the sum of ten cubes in exactly six ways.">[https://oeis.org/A345808 A345808]</span>||<span title="Numbers that are the sum of ten cubes in exactly seven ways.">[https://oeis.org/A345809 A345809]</span>||<span title="Numbers that are the sum of ten cubes in exactly eight ways.">[https://oeis.org/A345810 A345810]</span>||<span title="Numbers that are the sum of ten cubes in exactly nine ways.">[https://oeis.org/A345811 A345811]</span>||<span title="Numbers that are the sum of ten cubes in exactly ten ways.">[https://oeis.org/A345812 A345812]</span>
 
|-
 
|}
 
 
 
=== 4th powers ===
 
==== Numbers that can be expressed as the sum of k fourth powers in m or more ways (Table R4)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3!!m&gt;=4!!m&gt;=5!!m&gt;=6!!m&gt;=7!!m&gt;=8!!m&gt;=9!!m&gt;=10
 
|-
 
| k=1 ||<span title="Fourth powers: a(n) = n^4.">[https://oeis.org/A000583 A000583]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=3 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of three fourth powers in three or more ways.">[https://oeis.org/A344239 A344239]</span>||<span title="Numbers that are the sum of three fourth powers in four or more ways.">[https://oeis.org/A344277 A344277]</span>||<span title="Numbers that are the sum of three fourth powers in five or more ways.">[https://oeis.org/A344364 A344364]</span>||<span title="Numbers that are the sum of three fourth powers in six or more ways.">[https://oeis.org/A344647 A344647]</span>||<span title="Numbers that are the sum of three fourth powers in seven or more ways.">[https://oeis.org/A344729 A344729]</span>||<span title="Numbers that are the sum of three fourth powers in eight or more ways.">[https://oeis.org/A344737 A344737]</span>||<span title="Numbers that are the sum of three fourth powers in nine or more ways.">[https://oeis.org/A344750 A344750]</span>||<span title="Numbers that are the sum of three fourth powers in ten or more ways.">[https://oeis.org/A344862 A344862]</span>
 
|-
 
| k=4 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of four fourth powers in three or more ways.">[https://oeis.org/A344241 A344241]</span>||<span title="Numbers that are the sum of four fourth powers in four or more ways.">[https://oeis.org/A344352 A344352]</span>||<span title="Numbers that are the sum of four fourth powers in five or more ways.">[https://oeis.org/A344356 A344356]</span>||<span title="Numbers that are the sum of four fourth powers in six or more ways.">[https://oeis.org/A344904 A344904]</span>||<span title="Numbers that are the sum of four fourth powers in seven or more ways.">[https://oeis.org/A344922 A344922]</span>||<span title="Numbers that are the sum of four fourth powers in eight or more ways.">[https://oeis.org/A344924 A344924]</span>||<span title="Numbers that are the sum of four fourth powers in nine or more ways.">[https://oeis.org/A344926 A344926]</span>||<span title="Numbers that are the sum of four fourth powers in ten or more ways.">[https://oeis.org/A344928 A344928]</span>
 
|-
 
| k=5 ||&#xa0;||<span title="Numbers that are the sum of five fourth powers in two or more ways.">[https://oeis.org/A344238 A344238]</span>||<span title="Numbers that are the sum of five fourth powers in three or more ways.">[https://oeis.org/A344243 A344243]</span>||<span title="Numbers that are the sum of five fourth powers in four or more ways.">[https://oeis.org/A344354 A344354]</span>||<span title="Numbers that are the sum of five fourth powers in five or more ways.">[https://oeis.org/A344358 A344358]</span>||<span title="Numbers that are the sum of five fourth powers in six or more ways.">[https://oeis.org/A344940 A344940]</span>||<span title="Numbers that are the sum of five fourth powers in seven or more ways.">[https://oeis.org/A344942 A344942]</span>||<span title="Numbers that are the sum of five fourth powers in eight or more ways.">[https://oeis.org/A344944 A344944]</span>||<span title="Numbers that are the sum of five fourth powers in nine or more ways.">[https://oeis.org/A341891 A341891]</span>||<span title="Numbers that are the sum of five fourth powers in ten or more ways.">[https://oeis.org/A341897 A341897]</span>
 
|-
 
| k=6 ||&#xa0;||<span title="Numbers that are the sum of six fourth powers in two or more ways.">[https://oeis.org/A345559 A345559]</span>||<span title="Numbers that are the sum of six fourth powers in three or more ways.">[https://oeis.org/A345560 A345560]</span>||<span title="Numbers that are the sum of six fourth powers in four or more ways.">[https://oeis.org/A345561 A345561]</span>||<span title="Numbers that are the sum of six fourth powers in five or more ways.">[https://oeis.org/A345562 A345562]</span>||<span title="Numbers that are the sum of six fourth powers in six or more ways.">[https://oeis.org/A345563 A345563]</span>||<span title="Numbers that are the sum of six fourth powers in seven or more ways.">[https://oeis.org/A345564 A345564]</span>||<span title="Numbers that are the sum of six fourth powers in eight or more ways.">[https://oeis.org/A345565 A345565]</span>||<span title="Numbers that are the sum of six fourth powers in nine or more ways.">[https://oeis.org/A345566 A345566]</span>||<span title="Numbers that are the sum of six fourth powers in ten or more ways.">[https://oeis.org/A345567 A345567]</span>
 
|-
 
| k=7 ||&#xa0;||<span title="Numbers that are the sum of seven fourth powers in two or more ways.">[https://oeis.org/A345568 A345568]</span>||<span title="Numbers that are the sum of seven fourth powers in three or more ways.">[https://oeis.org/A345569 A345569]</span>||<span title="Numbers that are the sum of seven fourth powers in four or more ways.">[https://oeis.org/A345570 A345570]</span>||<span title="Numbers that are the sum of seven fourth powers in five or more ways.">[https://oeis.org/A345571 A345571]</span>||<span title="Numbers that are the sum of seven fourth powers in six or more ways.">[https://oeis.org/A345572 A345572]</span>||<span title="Numbers that are the sum of seven fourth powers in seven or more ways.">[https://oeis.org/A345573 A345573]</span>||<span title="Numbers that are the sum of seven fourth powers in eight or more ways.">[https://oeis.org/A345574 A345574]</span>||<span title="Numbers that are the sum of seven fourth powers in nine or more ways.">[https://oeis.org/A345575 A345575]</span>||<span title="Numbers that are the sum of seven fourth powers in ten or more ways.">[https://oeis.org/A345576 A345576]</span>
 
|-
 
| k=8 ||&#xa0;||<span title="Numbers that are the sum of eight fourth powers in two or more ways.">[https://oeis.org/A345577 A345577]</span>||<span title="Numbers that are the sum of eight fourth powers in three or more ways.">[https://oeis.org/A345578 A345578]</span>||<span title="Numbers that are the sum of eight fourth powers in four or more ways.">[https://oeis.org/A345579 A345579]</span>||<span title="Numbers that are the sum of eight fourth powers in five or more ways.">[https://oeis.org/A345580 A345580]</span>||<span title="Numbers that are the sum of eight fourth powers in six or more ways.">[https://oeis.org/A345581 A345581]</span>||<span title="Numbers that are the sum of eight fourth powers in seven or more ways.">[https://oeis.org/A345582 A345582]</span>||<span title="Numbers that are the sum of eight fourth powers in eight or more ways.">[https://oeis.org/A345583 A345583]</span>||<span title="Numbers that are the sum of eight fourth powers in nine or more ways.">[https://oeis.org/A345584 A345584]</span>||<span title="Numbers that are the sum of eight fourth powers in ten or more ways.">[https://oeis.org/A345585 A345585]</span>
 
|-
 
| k=9 ||&#xa0;||<span title="Numbers that are the sum of nine fourth powers in two or more ways.">[https://oeis.org/A345586 A345586]</span>||<span title="Numbers that are the sum of nine fourth powers in three or more ways.">[https://oeis.org/A345587 A345587]</span>||<span title="Numbers that are the sum of nine fourth powers in four or more ways.">[https://oeis.org/A345588 A345588]</span>||<span title="Numbers that are the sum of nine fourth powers in five or more ways.">[https://oeis.org/A345589 A345589]</span>||<span title="Numbers that are the sum of nine fourth powers in six or more ways.">[https://oeis.org/A345590 A345590]</span>||<span title="Numbers that are the sum of nine fourth powers in seven or more ways.">[https://oeis.org/A345591 A345591]</span>||<span title="Numbers that are the sum of nine fourth powers in eight or more ways.">[https://oeis.org/A345592 A345592]</span>||<span title="Numbers that are the sum of nine fourth powers in nine or more ways.">[https://oeis.org/A345593 A345593]</span>||<span title="Numbers that are the sum of nine fourth powers in ten or more ways.">[https://oeis.org/A345594 A345594]</span>
 
|-
 
| k=10 ||&#xa0;||<span title="Numbers that are the sum of ten fourth powers in two or more ways.">[https://oeis.org/A345595 A345595]</span>||<span title="Numbers that are the sum of ten fourth powers in three or more ways.">[https://oeis.org/A345596 A345596]</span>||<span title="Numbers that are the sum of ten fourth powers in four or more ways.">[https://oeis.org/A345597 A345597]</span>||<span title="Numbers that are the sum of ten fourth powers in five or more ways.">[https://oeis.org/A345598 A345598]</span>||<span title="Numbers that are the sum of ten fourth powers in six or more ways.">[https://oeis.org/A345599 A345599]</span>||<span title="Numbers that are the sum of ten fourth powers in seven or more ways.">[https://oeis.org/A345600 A345600]</span>||<span title="Numbers that are the sum of ten fourth powers in eight or more ways.">[https://oeis.org/A345601 A345601]</span>||<span title="Numbers that are the sum of ten fourth powers in nine or more ways.">[https://oeis.org/A345602 A345602]</span>||<span title="Numbers that are the sum of ten fourth powers in ten or more ways.">[https://oeis.org/A345603 A345603]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k fourth powers in exactly m ways (Table S4)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
 
|-
 
| k=2 ||<span title="Numbers that are the sum of two positive fourth powers in exactly one way.">[https://oeis.org/A344187 A344187]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=3 ||<span title="Numbers that are the sum of three fourth powers in exactly one way">[https://oeis.org/A344188 A344188]</span>||<span title="Numbers that are the sum of three fourth powers in exactly two ways.">[https://oeis.org/A344192 A344192]</span>||<span title="Numbers that are the sum of three fourth powers in exactly three ways.">[https://oeis.org/A344240 A344240]</span>||<span title="Numbers that are the sum of three fourth powers in exactly four ways.">[https://oeis.org/A344278 A344278]</span>||<span title="Numbers that are the sum of three fourth powers in exactly five ways.">[https://oeis.org/A344365 A344365]</span>||<span title="Numbers that are the sum of three fourth powers in exactly six ways.">[https://oeis.org/A344648 A344648]</span>||<span title="Numbers that are the sum of three fourth powers in exactly seven ways.">[https://oeis.org/A344730 A344730]</span>||<span title="Numbers that are the sum of three fourth powers in exactly eight ways.">[https://oeis.org/A344738 A344738]</span>||<span title="Numbers that are the sum of three fourth powers in exactly nine ways.">[https://oeis.org/A344751 A344751]</span>||<span title="Numbers that are the sum of three fourth powers in exactly ten ways.">[https://oeis.org/A344861 A344861]</span>
 
|-
 
| k=4 ||<span title="Numbers that are the sum of four fourth powers in exactly one way.">[https://oeis.org/A344189 A344189]</span>||<span title="Numbers that are the sum of four fourth powers in exactly two ways">[https://oeis.org/A344193 A344193]</span>||<span title="Numbers that are the sum of four fourth powers in exactly three ways.">[https://oeis.org/A344242 A344242]</span>||<span title="Numbers that are the sum of four fourth powers in exactly four ways.">[https://oeis.org/A344353 A344353]</span>||<span title="Numbers that are the sum of four fourth powers in exactly five ways.">[https://oeis.org/A344357 A344357]</span>||<span title="Numbers that are the sum of four fourth powers in exactly six ways.">[https://oeis.org/A344921 A344921]</span>||<span title="Numbers that are the sum of four fourth powers in exactly seven ways.">[https://oeis.org/A344923 A344923]</span>||<span title="Numbers that are the sum of four fourth powers in exactly eight ways.">[https://oeis.org/A344925 A344925]</span>||<span title="Numbers that are the sum of four fourth powers in exactly nine ways.">[https://oeis.org/A344927 A344927]</span>||<span title="Numbers that are the sum of four fourth powers in exactly ten ways.">[https://oeis.org/A344929 A344929]</span>
 
|-
 
| k=5 ||<span title="Numbers that are the sum of five fourth powers in exactly one way.">[https://oeis.org/A344190 A344190]</span>||<span title="Numbers that are the sum of five fourth powers in exactly two ways.">[https://oeis.org/A344237 A344237]</span>||<span title="Numbers that are the sum of five fourth powers in exactly three ways.">[https://oeis.org/A344244 A344244]</span>||<span title="Numbers that are the sum of five fourth powers in exactly four ways.">[https://oeis.org/A344355 A344355]</span>||<span title="Numbers that are the sum of five fourth powers in exactly five ways.">[https://oeis.org/A344359 A344359]</span>||<span title="Numbers that are the sum of five fourth powers in exactly six ways.">[https://oeis.org/A344941 A344941]</span>||<span title="Numbers that are the sum of five fourth powers in exactly seven ways.">[https://oeis.org/A344943 A344943]</span>||<span title="Numbers that are the sum of five fourth powers in exactly eight ways.">[https://oeis.org/A344945 A344945]</span>||<span title="Numbers that are the sum of five fourth powers in exactly nine ways.">[https://oeis.org/A341892 A341892]</span>||<span title="Numbers that are the sum of five fourth powers in exactly ten ways.">[https://oeis.org/A341898 A341898]</span>
 
|-
 
| k=6 ||<span title="Numbers that are the sum of six fourth powers in exactly one ways.">[https://oeis.org/A345813 A345813]</span>||<span title="Numbers that are the sum of six fourth powers in exactly two ways.">[https://oeis.org/A345814 A345814]</span>||<span title="Numbers that are the sum of six fourth powers in exactly three ways.">[https://oeis.org/A345815 A345815]</span>||<span title="Numbers that are the sum of six fourth powers in exactly four ways.">[https://oeis.org/A345816 A345816]</span>||<span title="Numbers that are the sum of six fourth powers in exactly five ways.">[https://oeis.org/A345817 A345817]</span>||<span title="Numbers that are the sum of six fourth powers in exactly six ways.">[https://oeis.org/A345818 A345818]</span>||<span title="Numbers that are the sum of six fourth powers in exactly seven ways.">[https://oeis.org/A345819 A345819]</span>||<span title="Numbers that are the sum of six fourth powers in exactly eight ways.">[https://oeis.org/A345820 A345820]</span>||<span title="Numbers that are the sum of six fourth powers in exactly nine ways.">[https://oeis.org/A345821 A345821]</span>||<span title="Numbers that are the sum of six fourth powers in exactly ten ways.">[https://oeis.org/A345822 A345822]</span>
 
|-
 
| k=7 ||<span title="Numbers that are the sum of seven fourth powers in exactly one ways.">[https://oeis.org/A345823 A345823]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly two ways.">[https://oeis.org/A345824 A345824]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly three ways.">[https://oeis.org/A345825 A345825]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly four ways.">[https://oeis.org/A345826 A345826]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly five ways.">[https://oeis.org/A345827 A345827]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly six ways.">[https://oeis.org/A345828 A345828]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly seven ways.">[https://oeis.org/A345829 A345829]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly eight ways.">[https://oeis.org/A345830 A345830]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly nine ways.">[https://oeis.org/A345831 A345831]</span>||<span title="Numbers that are the sum of seven fourth powers in exactly ten ways.">[https://oeis.org/A345832 A345832]</span>
 
|-
 
| k=8 ||<span title="Numbers that are the sum of eight fourth powers in exactly one ways.">[https://oeis.org/A345833 A345833]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly two ways.">[https://oeis.org/A345834 A345834]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly three ways.">[https://oeis.org/A345835 A345835]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly four ways.">[https://oeis.org/A345836 A345836]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly five ways.">[https://oeis.org/A345837 A345837]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly six ways.">[https://oeis.org/A345838 A345838]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly seven ways.">[https://oeis.org/A345839 A345839]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly eight ways.">[https://oeis.org/A345840 A345840]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly nine ways.">[https://oeis.org/A345841 A345841]</span>||<span title="Numbers that are the sum of eight fourth powers in exactly ten ways.">[https://oeis.org/A345842 A345842]</span>
 
|-
 
| k=9 ||<span title="Numbers that are the sum of nine fourth powers in exactly one ways.">[https://oeis.org/A345843 A345843]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly two ways.">[https://oeis.org/A345844 A345844]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly three ways.">[https://oeis.org/A345845 A345845]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly four ways.">[https://oeis.org/A345846 A345846]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly five ways.">[https://oeis.org/A345847 A345847]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly six ways.">[https://oeis.org/A345848 A345848]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly seven ways.">[https://oeis.org/A345849 A345849]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly eight ways.">[https://oeis.org/A345850 A345850]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly nine ways.">[https://oeis.org/A345851 A345851]</span>||<span title="Numbers that are the sum of nine fourth powers in exactly ten ways.">[https://oeis.org/A345852 A345852]</span>
 
|-
 
| k=10 ||<span title="Numbers that are the sum of ten fourth powers in exactly one ways.">[https://oeis.org/A345853 A345853]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly two ways.">[https://oeis.org/A345854 A345854]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly three ways.">[https://oeis.org/A345855 A345855]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly four ways.">[https://oeis.org/A345856 A345856]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly five ways.">[https://oeis.org/A345857 A345857]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly six ways.">[https://oeis.org/A345858 A345858]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly seven ways.">[https://oeis.org/A345859 A345859]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly eight ways.">[https://oeis.org/A345860 A345860]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly nine ways.">[https://oeis.org/A345861 A345861]</span>||<span title="Numbers that are the sum of ten fourth powers in exactly ten ways.">[https://oeis.org/A345862 A345862]</span>
 
|-
 
|}
 
 
 
=== 5th powers ===
 
==== Numbers that can be expressed as the sum of k fifth powers in m or more ways (Table R5)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3!!m&gt;=4!!m&gt;=5!!m&gt;=6!!m&gt;=7!!m&gt;=8!!m&gt;=9!!m&gt;=10
 
|-
 
| k=1 ||<span title="Fifth powers: a(n) = n^5.">[https://oeis.org/A000584 A000584]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=3 ||&#xa0;||<span title="Numbers that are the sum of three fifth powers in two or more ways.">[https://oeis.org/A345010 A345010]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=4 ||&#xa0;||<span title="Numbers that are the sum of four fifth powers in two or more ways.">[https://oeis.org/A344644 A344644]</span>||<span title="Numbers that are the sum of four fifth powers in three or more ways.">[https://oeis.org/A345337 A345337]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=5 ||&#xa0;||<span title="Numbers that are the sum of five fifth powers in two or more ways.">[https://oeis.org/A342685 A342685]</span>||<span title="Numbers that are the sum of five fifth powers in three or more ways.">[https://oeis.org/A342687 A342687]</span>||<span title="Numbers that are the sum of five fifth powers in four or more ways.">[https://oeis.org/A344518 A344518]</span>||<span title="Numbers that are the sum of five fifth powers in five or more ways.">[https://oeis.org/A345863 A345863]</span>||<span title="Numbers that are the sum of five fifth powers in six or more ways.">[https://oeis.org/A345864 A345864]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=6 ||&#xa0;||<span title="Numbers that are the sum of six fifth powers in two or more ways.">[https://oeis.org/A345507 A345507]</span>||<span title="Numbers that are the sum of six fifth powers in three or more ways.">[https://oeis.org/A345604 A345604]</span>||<span title="Numbers that are the sum of six fifth powers in four or more ways.">[https://oeis.org/A345718 A345718]</span>||<span title="Numbers that are the sum of six fifth powers in five or more ways.">[https://oeis.org/A345719 A345719]</span>||<span title="Numbers that are the sum of six fifth powers in six or more ways.">[https://oeis.org/A345720 A345720]</span>||<span title="Numbers that are the sum of six fifth powers in seven or more ways.">[https://oeis.org/A345721 A345721]</span>||<span title="Numbers that are the sum of six fifth powers in eight or more ways.">[https://oeis.org/A345722 A345722]</span>||<span title="Numbers that are the sum of six fifth powers in nine or more ways.">[https://oeis.org/A345723 A345723]</span>||<span title="Numbers that are the sum of six fifth powers in ten or more ways.">[https://oeis.org/A344196 A344196]</span>
 
|-
 
| k=7 ||&#xa0;||<span title="Numbers that are the sum of seven fifth powers in two or more ways.">[https://oeis.org/A345605 A345605]</span>||<span title="Numbers that are the sum of seven fifth powers in three or more ways.">[https://oeis.org/A345606 A345606]</span>||<span title="Numbers that are the sum of seven fifth powers in four or more ways.">[https://oeis.org/A345607 A345607]</span>||<span title="Numbers that are the sum of seven fifth powers in five or more ways.">[https://oeis.org/A345608 A345608]</span>||<span title="Numbers that are the sum of seven fifth powers in six or more ways.">[https://oeis.org/A345609 A345609]</span>||<span title="Numbers that are the sum of seven fifth powers in seven or more ways.">[https://oeis.org/A345629 A345629]</span>||<span title="Numbers that are the sum of seven fifth powers in eight or more ways.">[https://oeis.org/A345630 A345630]</span>||<span title="Numbers that are the sum of seven fifth powers in nine or more ways.">[https://oeis.org/A345631 A345631]</span>||<span title="Numbers that are the sum of seven fifth powers in ten or more ways.">[https://oeis.org/A345643 A345643]</span>
 
|-
 
| k=8 ||&#xa0;||<span title="Numbers that are the sum of eight fifth powers in two or more ways.">[https://oeis.org/A345610 A345610]</span>||<span title="Numbers that are the sum of eight fifth powers in three or more ways.">[https://oeis.org/A345611 A345611]</span>||<span title="Numbers that are the sum of eight fifth powers in four or more ways.">[https://oeis.org/A345612 A345612]</span>||<span title="Numbers that are the sum of eight fifth powers in five or more ways.">[https://oeis.org/A345613 A345613]</span>||<span title="Numbers that are the sum of eight fifth powers in six or more ways.">[https://oeis.org/A345614 A345614]</span>||<span title="Numbers that are the sum of eight fifth powers in seven or more ways.">[https://oeis.org/A345615 A345615]</span>||<span title="Numbers that are the sum of eight fifth powers in eight or more ways.">[https://oeis.org/A345616 A345616]</span>||<span title="Numbers that are the sum of eight fifth powers in nine or more ways.">[https://oeis.org/A345617 A345617]</span>||<span title="Numbers that are the sum of eight fifth powers in ten or more ways.">[https://oeis.org/A345618 A345618]</span>
 
|-
 
| k=9 ||&#xa0;||<span title="Numbers that are the sum of nine fifth powers in two or more ways.">[https://oeis.org/A345619 A345619]</span>||<span title="Numbers that are the sum of nine fifth powers in three or more ways.">[https://oeis.org/A345620 A345620]</span>||<span title="Numbers that are the sum of nine fifth powers in four or more ways.">[https://oeis.org/A345621 A345621]</span>||<span title="Numbers that are the sum of nine fifth powers in five or more ways.">[https://oeis.org/A345622 A345622]</span>||<span title="Numbers that are the sum of nine fifth powers in six or more ways.">[https://oeis.org/A345623 A345623]</span>||<span title="Numbers that are the sum of nine fifth powers in seven or more ways.">[https://oeis.org/A345624 A345624]</span>||<span title="Numbers that are the sum of nine fifth powers in eight or more ways.">[https://oeis.org/A345625 A345625]</span>||<span title="Numbers that are the sum of nine fifth powers in nine or more ways.">[https://oeis.org/A345626 A345626]</span>||<span title="Numbers that are the sum of nine fifth powers in ten or more ways.">[https://oeis.org/A345627 A345627]</span>
 
|-
 
| k=10 ||&#xa0;||<span title="Numbers that are the sum of ten fifth powers in two or more ways.">[https://oeis.org/A345634 A345634]</span>||<span title="Numbers that are the sum of ten fifth powers in three or more ways.">[https://oeis.org/A345635 A345635]</span>||<span title="Numbers that are the sum of ten fifth powers in four or more ways.">[https://oeis.org/A345636 A345636]</span>||<span title="Numbers that are the sum of ten fifth powers in five or more ways.">[https://oeis.org/A345637 A345637]</span>||<span title="Numbers that are the sum of ten fifth powers in six or more ways.">[https://oeis.org/A345638 A345638]</span>||<span title="Numbers that are the sum of ten fifth powers in seven or more ways.">[https://oeis.org/A345639 A345639]</span>||<span title="Numbers that are the sum of ten fifth powers in eight or more ways.">[https://oeis.org/A345640 A345640]</span>||<span title="Numbers that are the sum of ten fifth powers in nine or more ways.">[https://oeis.org/A345641 A345641]</span>||<span title="Numbers that are the sum of ten fifth powers in ten or more ways.">[https://oeis.org/A345642 A345642]</span>
 
|-
 
|}
 
==== Numbers that can be expressed as the sum of k fifth powers in exactly m ways (Table S5)====
 
{| class="wikitable" style="text-align:left"
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
 
|-
 
| k=3 ||<span title="Numbers that are the sum of three fifth powers in exactly one way.">[https://oeis.org/A344641 A344641]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=4 ||<span title="Numbers that are the sum of four fifth powers in exactly one way.">[https://oeis.org/A344642 A344642]</span>||<span title="Numbers that are the sum of four fifth powers in exactly two ways.">[https://oeis.org/A344645 A344645]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=5 ||<span title="Numbers that are the sum of five fifth powers in exactly one way.">[https://oeis.org/A344643 A344643]</span>||<span title="Numbers that are the sum of five fifth powers in exactly two ways.">[https://oeis.org/A342686 A342686]</span>||<span title="Numbers that are the sum of five fifth powers in exactly three ways.">[https://oeis.org/A342688 A342688]</span>||<span title="Numbers that are the sum of five fifth powers in exactly four ways.">[https://oeis.org/A344519 A344519]</span>||<span title="Numbers that are the sum of five fifth powers in exactly five ways.">[https://oeis.org/A346257 A346257]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
| k=6 ||<span title="Numbers that are the sum of six fifth powers in exactly one way.">[https://oeis.org/A346356 A346356]</span>||<span title="Numbers that are the sum of six fifth powers in exactly two ways.">[https://oeis.org/A346357 A346357]</span>||<span title="Numbers that are the sum of six fifth powers in exactly three ways.">[https://oeis.org/A346358 A346358]</span>||<span title="Numbers that are the sum of six fifth powers in exactly four ways.">[https://oeis.org/A346359 A346359]</span>||<span title="Numbers that are the sum of six fifth powers in exactly five ways.">[https://oeis.org/A346360 A346360]</span>||<span title="Numbers that are the sum of six fifth powers in exactly six ways.">[https://oeis.org/A346361 A346361]</span>||<span title="Numbers that are the sum of six fifth powers in exactly seven ways.">[https://oeis.org/A346362 A346362]</span>||<span title="Numbers that are the sum of six fifth powers in exactly eight ways.">[https://oeis.org/A346363 A346363]</span>||<span title="Numbers that are the sum of six fifth powers in exactly nine ways.">[https://oeis.org/A346364 A346364]</span>||<span title="Numbers that are the sum of six fifth powers in exactly ten ways.">[https://oeis.org/A346365 A346365]</span>
 
|-
 
| k=7 ||<span title="Numbers that are the sum of seven fifth powers in exactly one way.">[https://oeis.org/A346278 A346278]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly two ways.">[https://oeis.org/A346279 A346279]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly three ways.">[https://oeis.org/A346280 A346280]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly four ways.">[https://oeis.org/A346281 A346281]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly five ways.">[https://oeis.org/A346282 A346282]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly six ways.">[https://oeis.org/A346283 A346283]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly seven ways.">[https://oeis.org/A346284 A346284]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly eight ways.">[https://oeis.org/A346285 A346285]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly nine ways.">[https://oeis.org/A346286 A346286]</span>||<span title="Numbers that are the sum of seven fifth powers in exactly ten ways.">[https://oeis.org/A346259 A346259]</span>
 
|-
 
| k=8 ||<span title="Numbers that are the sum of eight fifth powers in exactly one way.">[https://oeis.org/A346326 A346326]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly two ways.">[https://oeis.org/A346327 A346327]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly three ways.">[https://oeis.org/A346328 A346328]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly four ways.">[https://oeis.org/A346329 A346329]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly five ways.">[https://oeis.org/A346330 A346330]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly six ways.">[https://oeis.org/A346331 A346331]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly seven ways.">[https://oeis.org/A346332 A346332]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly eight ways.">[https://oeis.org/A346333 A346333]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly nine ways.">[https://oeis.org/A346334 A346334]</span>||<span title="Numbers that are the sum of eight fifth powers in exactly ten ways.">[https://oeis.org/A346335 A346335]</span>
 
|-
 
| k=9 ||<span title="Numbers that are the sum of nine fifth powers in exactly one way.">[https://oeis.org/A346336 A346336]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly two ways.">[https://oeis.org/A346337 A346337]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly three ways.">[https://oeis.org/A346338 A346338]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly four ways.">[https://oeis.org/A346339 A346339]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly five ways.">[https://oeis.org/A346340 A346340]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly six ways.">[https://oeis.org/A346341 A346341]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly seven ways.">[https://oeis.org/A346342 A346342]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly eight ways.">[https://oeis.org/A346343 A346343]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly nine ways.">[https://oeis.org/A346344 A346344]</span>||<span title="Numbers that are the sum of nine fifth powers in exactly ten ways.">[https://oeis.org/A346345 A346345]</span>
 
|-
 
| k=10 ||<span title="Numbers that are the sum of ten fifth powers in exactly one way.">[https://oeis.org/A346346 A346346]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly two ways.">[https://oeis.org/A346347 A346347]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly three ways.">[https://oeis.org/A346348 A346348]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly four ways.">[https://oeis.org/A346349 A346349]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly five ways.">[https://oeis.org/A346350 A346350]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly six ways.">[https://oeis.org/A346351 A346351]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly seven ways.">[https://oeis.org/A346352 A346352]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly eight ways.">[https://oeis.org/A346353 A346353]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly nine ways.">[https://oeis.org/A346354 A346354]</span>||<span title="Numbers that are the sum of ten fifth powers in exactly ten ways.">[https://oeis.org/A346355 A346355]</span>
 
|-
 
|}
 

Revision as of 15:49, 5 March 2022

A055277 Triangle T(n,k) of number of rooted trees with n nodes and k leaves, 1 <= k <= n.

The triangle starts:

      A002 A550 A550 A550 A550 A550 A550 A550 A550 A550 A550  
       620  278  279  280  281  282  283  284  285  286  287
n/k 1    2    3    4    5    6    7    8    9   10   11   12
--+---------------------------------------------------------------
 1| 1
 2| 1    0
 3| 1    1    0
 4| 1    2    1    0
 5| 1    4    3    1    0
 6| 1    6    8    4    1    0
 7| 1    9   18   14    5    1    0
 8| 1   12   35   39   21    6    1    0
 9| 1   16   62   97   72   30    7    1    0
10| 1   20  103  212  214  120   40    8    1    0
11| 1   25  161  429  563  416  185   52    9    1    0
12| 1   30  241  804 1344 1268  732  270   65   10    1    0
13| 1   36  348 1427 2958 3499 2544 1203  378   80   11    1    0

For some columns, an o.g.f. is already given or conjectured:

A002620 (column 2): x^2 / ((1+x)*(1-x)^3)
A055278 (column 3): x^4*(x^3+x^2+1) / ((1-x^2)*(1-x^3)*(1-x)^3)
A055279 (column 4): x^5*(1+x+3*x^2+5*x^3+7*x^4+5*x^5+5*x^6+2*x^7+x^8) / ((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4))

When o.g.f.s for higher columns are "guessed" (with Maple's guessgf), it turns out that the denominators are all products of cyclotomic polynomials. The following factorizations of the denominators are obtained:

deo[2 ] :=                                                                                             (x^2-1)  *(x-1)^2;
deo[3 ] :=                                                                                   (x^3-1)  *(x^2-1)  *(x-1)^3;
deo[4 ] :=                                                                         (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^3;
deo[5 ] :=                                                               (x^5-1)  *(x^4-1)            *(x^2-1)^2*(x-1)^5;
deo[6 ] :=                                                     (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^3;
deo[7 ] :=                                             (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^4;
deo[8 ] :=                                     (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)  *(x^2-1)^3*(x-1)^5;
deo[9 ] :=                             (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^3*(x-1)^4;
deo[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^3;
deo[11] :=           (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^5;
deo[12] :=  (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^3*(x^2-1)^3*(x-1)^4;

Some pattern can be seen, but the higher exponents are not systematic. The problem is that some factors may have been cancelled out in the o.g.f.s polynomial fraction. However, by multiplying both the nominator and the denominator polynomials with some factor, the following form of the denominators can be obtained:

den[2 ] :=                                                                                             (x^2-1)  *(x-1)^2 ;
den[3 ] :=                                                                                   (x^3-1)  *(x^2-1)  *(x-1)^3 ;
den[4 ] :=                                                                         (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^4 ;
den[5 ] :=                                                               (x^5-1)  *(x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^5 ;
den[6 ] :=                                                     (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^6 ;
den[7 ] :=                                             (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^7 ;
den[8 ] :=                                     (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^8 ;
den[9 ] :=                             (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^9 ;
den[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^5*(x-1)^10;
den[11] :=           (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^5*(x-1)^11;
den[12] :=  (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^4*(x^2-1)^6*(x-1)^12;

Here, the exponents increase in groups of 1 for (x-1), in groups of 2 for (x^2-1), in groups of 3 for (x^3-1), etc. The triangle of the exponents turns out to be A010766 (with rows reversed). The latter has a very simple formula T(n,k) = floor(n/k) and starts:

  n/k   1  2  3  4  5  6  7  8  9 10 11 12
  ---+------------------------------------
   1 |  1
   2 |  2  1
   3 |  3  1  1
   4 |  4  2  1  1
   5 |  5  2  1  1  1
   6 |  6  3  2  1  1  1
   7 |  7  3  2  1  1  1  1
   8 |  8  4  2  2  1  1  1  1
   9 |  9  4  3  2  1  1  1  1  1
  10 | 10  5  3  2  2  1  1  1  1  1
  11 | 11  5  3  2  2  1  1  1  1  1  1
  12 | 12  6  4  3  2  2  1  1  1  1  1  1

In order to support this hypothesis, the following steps were performed:

  1. Determine the denomiator polynomials den above, and take their coefficients as the signature for a linear recurrence with constant coefficients.
  2. Compute a sufficient number of terms (150 - 270) for the column sequences A055278-A055287 with Michale Somos' PARI program in A055277. This took about 1 day.
  3. Take degree(den[n]) initial terms from step 2, run the recurrence and verify it for all terms.

In fact, the initial terms could be prefixed by n zeros.