Difference between revisions of "Talk:Main Page"

From teherba.org
Jump to: navigation, search
(Current, without A0033xx)
Line 1: Line 1:
=== <code>b[n]q</code>: rebase from base b into base q ===
+
=== Sums of like powers===
In 2005, [[User:Marc_LeBrun|Marc LeBrun]] described the rebasing notation (cf. A000695):
+
==== Sums of k m-th powers &gt;= 0 (List A)====
 
 
:This may be described concisely using the "rebase" notation <code>b[n]q</code>, which means "replace b with q in the expansion of n", thus rebasing" n from base b into base q. The present sequence is <code>2[n]4</code>. Many interesting operations (e.g., <code>10[n](1/10)</code> = digit reverse, shifted) are nicely expressible this way.
 
 
 
:Note that <code>q[n]b</code> is (roughly) inverse to <code>b[n]q</code>.
 
 
 
:It's also natural to generalize the idea of "basis" so as to cover the likes of <code>F[n]2</code>, the so-called "fibbinary" numbers (A003714) nd provide standard ready-made images of entities obeying other arithmetics, say like <code>GF2[n]2</code> (e.g., primes = A014580, squares = the present sequence, etc.).
 
 
 
The following table shows relevant pertinent sequences in the OEIS:
 
 
{| class="wikitable" style="text-align:left"
 
{| class="wikitable" style="text-align:left"
! &#xa0; !!b=2!!b=3!!b=4!!b=5!!b=6!!b=7!!b=8!!b=9!!b=10
+
! &#xa0; !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!&#xa0;!!&#xa0;!!m=13
 
|-
 
|-
| q=2 ||<center>&mdash;</center>||<span title="Rebase n from 3 to 2. Replace 3^k with 2^k in ternary expansion of n.">A065361</span>||<span title="Rebase n from 4 to 2. Replace 4^k with 2^k in quaternary expansion of n.">A065362<sup>2</sup></span>||<span title="a(n)=Sum{d(i)*2^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A215088</span>||<span title="a(n)=Sum{d(i)*6^i: i=0,1,...,m}, where Sum{d(i)*2^i: i=0,1,...,m} is the base 2 representation of n.">A215089</span>||<span title="a(n) = Sum{d(i)*2^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m}=n, d(i)∈{0,1,...,6}">A203580</span>||&#xa0;||&#xa0;||<span title="If n = Sum c_i 10^i then a(n) = Sum c_i 2^i.">A028897</span>
+
| 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;
 
|-
 
|-
| q=3 ||<span title="Numbers n whose base 3 representation contains no 2.">A005836<sup>1</sup></span>||<center>&mdash;</center>||<span title="a(n) = Sum_{i=0..m} d(i)*3^i, where Sum_{i=0..m} d(i)*4^i is the base-4 representation of n.">A215090</span>||&#xa0;||<span title="a(n) = Sum_{i=0..m} d(i)*3^i, where Sum_{i=0..m} d(i)*6^i is the base-6 representation of n.">A215092<sup>2</sup></span>||&#xa0;||&#xa0;||&#xa0;||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 3^i.">A028898</span>
+
| 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>
 
|-
 
|-
| q=4 ||<span title="Moser-de Bruijn sequence: sums of distinct powers of 4.">A000695</span>||<span title="Numbers with no 3''s in base-4 expansion.">A023717</span>||<center>&mdash;</center>||<span title="a(n) = Sum_{i=0..m} d(i)*4^i, where Sum_{i=0..m} d(i)*5^i is the base-5 representation of n.">A303787</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 4^i.">A028899</span>
+
| 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;
 
|-
 
|-
| q=5 ||<span title="Sums of distinct powers of 5.">A033042</span>||<span title="Positive numbers n such that the base 5 representation of n contains no 3 or 4.">A037453<sup>2</sup></span>||<span title="Sum{d(i)*5^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037459<sup>2</sup></span>||<center>&mdash;</center>||<span title="a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*6^i is the base-6 representation of n.">A303788</span>||&#xa0;||&#xa0;||&#xa0;||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 5^i.">A028900</span>
+
| 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;
 
|-
 
|-
| q=6 ||<span title="Sums of distinct powers of 6.">A033043</span>||<span title="a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*3^i is the base 3 representation of n.">A037454</span>||<span title="a(n)=Sum{d(i)*6^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037460</span>||<span title="Sum{d(i)*6^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037465</span>||<center>&mdash;</center>||<span title="a(n) = Sum_{i=0..m} d(i)*6^i, where Sum_{i=0..m} d(i)*7^i is the base-7 representation of n.">A303789</span>||&#xa0;||&#xa0;||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 6^i.">A028901</span>
+
| 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;
 
|-
 
|-
| q=7 ||<span title="Sums of distinct powers of 7.">A033044<sup>1</sup></span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*3^i: i=0,1,...,m} is the base 3 representation of n.">A037455</span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037461<sup>2</sup></span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037466</span>||<span title="a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.">A037470</span>||<center>&mdash;</center>||&#xa0;||&#xa0;||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 7^i.">A028902</span>
+
|}
 +
==== Sums of exactly k positive m-th powers &gt; 0 (List B)====
 +
{| class="wikitable" style="text-align:left"
 +
! &#xa0; !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
 
|-
 
|-
| q=8 ||<span title="Sums of distinct powers of 8.">A033045</span>||<span title="a(n)=Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*3^i: i=0,1,...,m} is the base 3 representation of n.">A037456</span>||<span title="a(n) = Sum_{i = 0..m} d(i)*8^i, where Sum_{i = 0..m} d(i)*4^i is the base 4 representation of n.">A037462</span>||<span title="Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037467</span>||<span title="a(n)=Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.">A037471</span>||<span title="a(n) = Sum{d(i)*8^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.">A037474</span>||<center>&mdash;</center>||&#xa0;||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 8^i.">A028903</span>
+
| 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>
 
|-
 
|-
| q=9 ||<span title="Sums of distinct powers of 9.">A033046</span>||&#xa0;||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.">A037463</span>||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.">A037468</span>||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.">A037472</span>||<span title="a(n)=Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.">A037475</span>||<span title="a(n) = Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*8^i: i=0,1,...,m} is the base 8 representation of n.">A037477</span>||<center>&mdash;</center>||<span title="Map n = Sum c_i 10^i to a(n) = Sum c_i 9^i.">A028904</span>
+
| 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>
 
|-
 
|-
| q=10 ||<span title="The binary numbers (or binary words, or binary vectors): numbers written in base 2.">A007088</span>||<span title="Numbers in base 3.">A007089</span>||<span title="Numbers in base 4.">A007090</span>||<span title="Numbers in base 5.">A007091</span>||<span title="Numbers in base 6.">A007092</span>||<span title="Numbers in base 7.">A007093</span>||<span title="Numbers in base 8.">A007094</span>||<span title=" Numbers in base 9.">A007095</span><br><span title="a(n)=Sum{d(i)*10^i: i=0,1,...,m}, where Sum{d(i)*9^i: i=0,1,...,m} is the base 9 representation of n.">A037479<sup>2</sup></span>||<center>&mdash;</center>
+
| 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>
 
|-
 
|-
| q=11 ||<span title="Sums of distinct powers of 11.">A033047</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
| 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>
 
|-
 
|-
| q=12 ||<span title="Sums of distinct powers of 12.">A033048</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers in base-12 representation that can be written with decimal digits.">A102487<sup>1</sup></span>
+
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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;
 +
|-
 +
| 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;
 +
|-
 +
|}
 +
==== Sums of at most k positive m-th powers &gt; 0 (List C)====
 +
{| class="wikitable" style="text-align:left"
 +
! &#xa0; !!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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>
 +
|-
 +
| 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 (List 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 (List 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===
 +
==== Numbers that can be expressed as the sum of k squares in m or more ways (List R2)====
 +
{| 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;||<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 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>
 +
|-
 +
| 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 squares in exactly m ways (List S2)====
 +
{| 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 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025302 A025302]</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 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 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</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>||<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 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>||&#xa0;
 +
|-
 +
| 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>||&#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;
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k cubes in m or more ways (List R3)====
 +
{| 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 cubes: a(n) = n^3.">[https://oeis.org/A000578 A000578]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#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;||<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;||<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>||<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 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>||<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 cubes in exactly m ways (List S3)====
 +
{| 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 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>||<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 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>||<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>
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k fourth powers in m or more ways (List 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 (List 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>
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k fifth powers in m or more ways (List 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 (List 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
 
|-
 
|-
| q=13 ||<span title="Sums of distinct powers of 13.">A033049</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||<span title="If n = c0 + c1*10 + c2*10^2 + ...cn*10^n then a(n) = c0 + c1*13 + c2*13^2 + ...cn*13^k.">A094823</span>
+
| 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;
 
|-
 
|-
| q=14 ||<span title="Numbers whose set of base 14 digits is {0,1}.">A033050</span>||&#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;
 
|-
 
|-
| q=15 ||<span title="Numbers whose set of base 15 digits is {0,1}.">A033051</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;
 
|-
 
|-
| q=16 ||<span title="a(1) = 1, a(2n) = 16a(n), a(2n+1) = a(2n)+1.">A033052</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||<span title="Take the decimal representation of n and read it as if it were written in hexadecimal.">A102489</span>
+
| 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>
 
|-
 
|-
| q=17 ||<span title="a(0)=0, a(1)=1, a(2n)=17*a(n), a(2n+1)=a(2n)+1.">A197351</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
| 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>
 
|-
 
|-
| q=18 ||<span title="a(0)=0, a(1)=1, a(2n)=18*a(n), a(2n+1)=a(2n)+1.">A197352</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
| 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>
 
|-
 
|-
| q=19 ||<span title="a(0)=0, a(1)=1, a(2n)=19*a(n), a(2n+1)=a(2n)+1.">A197353</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
+
| 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>
 
|-
 
|-
| q=20 ||<span title="Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.">A063012</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;||<span title="Numbers whose base-20 representation can be written with decimal digits.">A102491<sup>1</sup></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>
 
|-
 
|-
 
|}
 
|}
<sup>1</sup> These sequences have offset 1 and start with n=0.<br>
 
<sup>2</sup> These sequences have offset 1 and start with n=1.<br>
 
All other sequences have offset 0 and start with n=0.<br>
 
===Sums of distinct powers of q===
 
The first column (b=2) of the table above shows the sequences for ''Sums of distinct powers of q'', since the binary digits in n enumerate all such powers.
 
===Examples===
 
A037454: 3[n]6
 
n =    0  1  2  3  4  5  6  7  8  9  10  11
 
a(n) = 0, 1, 2, 6, 7, 8, 12, 13, 14, 36, 37, 38, ...
 
n = 11: 11<sub>10</sub> = 102<sub>3</sub> -&gt; 102<sub>6</sub> = 1*6^2 + 0*6^1 + 2*6^0 = 38<sub>10</sub> = a(11)
 
===Programs===
 
* (Mathematica)
 
b:=3; q:=6; Table[FromDigits[RealDigits[n, b], q], {n, 0, 100}]
 
* (PARI)
 
b=3; q=6; for(n=0,100,print1(fromdigits(digits(n, b), q),","));
 
* Java ([https://github.com/archmageirvine/joeis jOEIS])
 
java -cp joeis.jar [https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a037/A037454.java irvine.oeis.a037.A037454] 3 6
 

Revision as of 20:40, 19 August 2021

Sums of like powers

Sums of k m-th powers >= 0 (List A)

  m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10     m=13
k>=2 A176209                      
k=-1 A336448 A294287 A294288 A294300 A294301 A294302 A155468 A007487 A294305     A181134
k=2 A140328 A004999 A018786                  
k=3 A294713 A332201 A193244                  
k=4   A001245                    

Sums of exactly k positive m-th powers > 0 (List B)

  m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11
k=2 A024509 A003325 A003336 A003347 A003358 A003369 A003380 A003391 A004802 A004813
k=3 A024795 A024981 A309762 A003348 A003359 A003370 A003381 A003392 A004803 A004814
k=4 A000414   A309763 A003349 A003360 A003371 A003382 A003393 A004804 A004815
k=5 A047700 A003328 A003339 A003350 A003361 A003372 A003383 A003394 A004805 A004816
k=6   A003329 A003340 A003351 A003362 A003373 A003384 A003395 A004806 A004817
k=7   A003330 A003341 A003352 A003363 A003374 A003385 A003396 A004807 A004818
k=8   A003331 A003342 A003353 A003364 A003375 A003386 A003397 A004808 A004819
k=9   A003332 A003343 A003354 A003365 A003376 A003387 A003398 A004809 A004820
k=10   A003333 A003344 A003355 A003366 A003377 A003388 A003399 A004810 A004821
k=11   A003334 A003345 A003356 A003367 A003378 A003389 A004800 A004811 A004822
k=12   A003335 A003346 A003357 A003368 A003379 A003390 A004801 A004812 A004823
k=13     A047724 A123294            
k=14     A047725 A123295            

Sums of at most k positive m-th powers > 0 (List C)

  m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11
k<=2   A004831 A004842 A004853 A004864 A004875 A004886 A004897 A004908
k<=3 A004825 A004832 A004843 A004854 A004865 A004876 A004887 A004898 A004909
k<=4 A004826 A004833 A004844 A004855 A004866 A004877 A004888 A004899 A004910
k<=5 A004827 A004834 A004845 A004856 A004867 A004878 A004889 A004900 A004911
k<=6 A004828 A004835 A004846 A004857 A004868 A004879 A004890 A004901 A004912
k<=7 A004829 A004836 A004847 A004858 A004869 A004880 A004891 A004902 A004913
k<=8 A004830 A004837 A004848 A004859 A004870 A004881 A004892 A004903 A004914
k<=9   A004838 A004849 A004860 A004871 A004882 A004893 A004904 A004915
k<=10   A004839 A004850 A004861 A004872 A004883 A004894 A004905 A004916
k<=11   A004840 A004851 A004862 A004873 A004884 A004895 A004906 A004917
k<=12   A004841 A004852 A004863 A004874 A004885 A004896 A004907 A004918

Sums of k positive m-th powers > 1 (List D)

  m=2 m=3
k=-1   A078131
k=2   A294073
k=3 A302359 A302360

Numbers that have exactly k representations as the sum of m squares >= 0 (List E)

  m=2     m=5 m=6 m=7
k=1         A295484  
k=2 A085625     A295150 A295485 A295742
k=3 A000443     A295151 A295486 A295743
k=4       A295152 A295487 A295744
k=5 A294716     A295153 A295488 A295745
k=6       A295154 A295489 A295747
k=7       A295155 A295490 A295748
k=8       A295156 A295491 A295749
k=9       A295157 A295492 A295750
k=10       A295158 A295493 A295751

New Lists

Numbers that can be expressed as the sum of k squares in m or more ways (List R2)

  m>=1 m>=2 m>=3 m>=4 m>=5 m>=6 m>=7 m>=8 m>=9 m>=10
k=1 A000290                  
k=2   A007692 A025313 A025314 A025315 A025316 A025317 A025318 A025319 A025320
k=3 A024796 A024804 A025349 A025350 A025351 A025352 A025353 A025354 A025355 A025356
k=4   A259058 A025387 A025388 A025389 A025390 A025391 A025392 A025393 A025394
k=5   A344795 A344796 A344797 A344798 A344799 A344800 A344801 A344802 A344803
k=6 A344805 A344806 A344807 A344808 A344809 A344810 A344811 A344812 A345476 A345477
k=7 A345478 A345479 A345480 A345481 A345482 A345483 A345484 A345485 A345486 A345487
k=8 A345488 A345489 A345490 A345491 A345492 A345493 A345494 A345495 A345496 A345497
k=9 A345498 A345499 A345500 A345501 A345502 A345503 A345504 A345505   A346803
k=10 A345508 A345509 A345510   A346804 A346805 A346806 A346807   A346808

Numbers that can be expressed as the sum of k squares in exactly m ways (List S2)

  m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11
k=2 A025302 A085625 A025304 A025305 A294716 A025307 A025308 A025309 A025310 A025311 A236711
k=3 A025339 A025340 A025341 A025342 A025343 A025344 A025345 A025346 A025347 A025348  
k=4 A025376 A025377 A025378 A025379 A025380 A025381 A025382 A025383 A025384 A025385  
k=5 A294675 A295150 A295151 A295152 A295153 A295154 A295155 A295156 A295157 A295158  
k=6 A295670 A295692 A295693 A295694 A295695 A295696 A295697 A295698 A295699 A295700  
k=7 A295797 A295799 A295800 A295801 A295802 A295803 A295804 A295805 A295806 A295807  

Numbers that can be expressed as the sum of k cubes in m or more ways (List R3)

  m>=1 m>=2 m>=3 m>=4 m>=5 m>=6 m>=7 m>=8 m>=9 m>=10
k=1 A000578                  
k=2 A001235   A018787 A023051 A051167          
k=3   A024974 A025402 A343968 A343967 A345083 A345086 A345087 A345119 A345121
k=4 A003327 A259060 A025413 A343971 A343987 A345148 A345150 A345152 A345146 A345155
k=5   A343702 A343704 A344034 A343989 A345174 A345180 A345183 A345185 A345187
k=6   A345511 A345512 A345513 A345514 A345515 A345516 A345517 A345518 A345519
k=7   A345520 A345521 A345522 A345523 A345524 A345525 A345526 A345527 A345506
k=8   A345532 A345533 A345534 A345535 A345536 A345537 A345538 A345539 A345540
k=9   A345541 A345542 A345543 A345544 A345545 A345546 A345547 A345548 A345549
k=10   A345550 A345551 A345552 A345553 A345554 A345555 A345556 A345557 A345558

Numbers that can be expressed as the sum of k cubes in exactly m ways (List S3)

  m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10
k=2 A338667 A343708 A344804 A345865            
k=3 A025399 A025400 A025401 A343969 A343970 A345084 A345085 A345088 A345120 A345122
k=4 A025408 A025409 A025410 A343972 A343988 A345149 A345151 A345153 A345154 A345156
k=5 A048926 A048927 A343705 A344035   A345175 A345181 A345184 A345186 A345188
k=6 A048929 A048930 A048931 A345766 A345767 A345768 A345769 A345770 A345771 A345772
k=7 A345773 A345774 A345775 A345776 A345777 A345778 A345779 A345780 A345781 A345782
k=8 A345783 A345784 A345785 A345786 A345787 A345788 A345789 A345790 A345791 A345792
k=9 A345793 A345794 A345795 A345796 A345797 A345798 A345799 A345800 A345801 A345802
k=10 A345803 A345804 A345805 A345806 A345807 A345808 A345809 A345810 A345811 A345812

Numbers that can be expressed as the sum of k fourth powers in m or more ways (List R4)

  m>=1 m>=2 m>=3 m>=4 m>=5 m>=6 m>=7 m>=8 m>=9 m>=10
k=1 A000583                  
k=3     A344239 A344277 A344364 A344647 A344729 A344737 A344750 A344862
k=4     A344241 A344352 A344356 A344904 A344922 A344924 A344926 A344928
k=5   A344238 A344243 A344354 A344358 A344940 A344942 A344944 A341891 A341897
k=6   A345559 A345560 A345561 A345562 A345563 A345564 A345565 A345566 A345567
k=7   A345568 A345569 A345570 A345571 A345572 A345573 A345574 A345575 A345576
k=8   A345577 A345578 A345579 A345580 A345581 A345582 A345583 A345584 A345585
k=9   A345586 A345587 A345588 A345589 A345590 A345591 A345592 A345593 A345594
k=10   A345595 A345596 A345597 A345598 A345599 A345600 A345601 A345602 A345603

Numbers that can be expressed as the sum of k fourth powers in exactly m ways (List S4)

  m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10
k=2 A344187                  
k=3 A344188 A344192 A344240 A344278 A344365 A344648 A344730 A344738 A344751 A344861
k=4 A344189 A344193 A344242 A344353 A344357 A344921 A344923 A344925 A344927 A344929
k=5 A344190 A344237 A344244 A344355 A344359 A344941 A344943 A344945 A341892 A341898
k=6 A345813 A345814 A345815 A345816 A345817 A345818 A345819 A345820 A345821 A345822
k=7 A345823 A345824 A345825 A345826 A345827 A345828 A345829 A345830 A345831 A345832
k=8 A345833 A345834 A345835 A345836 A345837 A345838 A345839 A345840 A345841 A345842
k=9 A345843 A345844 A345845 A345846 A345847 A345848 A345849 A345850 A345851 A345852
k=10 A345853 A345854 A345855 A345856 A345857 A345858 A345859 A345860 A345861 A345862

Numbers that can be expressed as the sum of k fifth powers in m or more ways (List R5)

  m>=1 m>=2 m>=3 m>=4 m>=5 m>=6 m>=7 m>=8 m>=9 m>=10
k=1 A000584                  
k=3   A345010                
k=4   A344644 A345337              
k=5   A342685 A342687 A344518 A345863 A345864        
k=6   A345507 A345604 A345718 A345719 A345720 A345721 A345722 A345723 A344196
k=7   A345605 A345606 A345607 A345608 A345609 A345629 A345630 A345631 A345643
k=8   A345610 A345611 A345612 A345613 A345614 A345615 A345616 A345617 A345618
k=9   A345619 A345620 A345621 A345622 A345623 A345624 A345625 A345626 A345627
k=10   A345634 A345635 A345636 A345637 A345638 A345639 A345640 A345641 A345642

Numbers that can be expressed as the sum of k fifth powers in exactly m ways (List S5)

  m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10
k=3 A344641                  
k=4 A344642 A344645                
k=5 A344643 A342686 A342688 A344519 A346257          
k=6 A346356 A346357 A346358 A346359 A346360 A346361 A346362 A346363 A346364 A346365
k=7 A346278 A346279 A346280 A346281 A346282 A346283 A346284 A346285 A346286 A346259
k=8 A346326 A346327 A346328 A346329 A346330 A346331 A346332 A346333 A346334 A346335
k=9 A346336 A346337 A346338 A346339 A346340 A346341 A346342 A346343 A346344 A346345
k=10 A346346 A346347 A346348 A346349 A346350 A346351 A346352 A346353 A346354 A346355