Difference between revisions of "Talk:Main Page"

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(Current, without A0033xx)
(squares and cubes (non)dist)
 
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=== Sums of like powers===
 
=== Sums of like powers===
==== Sums of k m-th powers >= 0 (List A)====
+
==== Sums of k m-th powers >= 0 (Table A)====
 
{| class="wikitable" style="text-align:left"
 
{| class="wikitable" style="text-align:left"
 
!   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!! !! !!m=13
 
!   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!! !! !!m=13
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|-
 
|-
 
|}
 
|}
==== Sums of exactly k positive m-th powers > 0 (List B)====
+
==== Sums of exactly k positive m-th powers > 0 (Table B)====
 
{| class="wikitable" style="text-align:left"
 
{| class="wikitable" style="text-align:left"
 
!   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
 
!   !!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
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|-
 
|-
 
|}
 
|}
==== Sums of at most k positive m-th powers > 0 (List C)====
+
==== Sums of at most k positive m-th powers > 0 (Table C)====
 
{| class="wikitable" style="text-align:left"
 
{| class="wikitable" style="text-align:left"
 
!   !!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
 
!   !!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11
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|-
 
|-
 
|}
 
|}
==== Sums of k positive m-th powers > 1 (List D)====
+
==== Sums of k positive m-th powers > 1 (Table D)====
 
{| class="wikitable" style="text-align:left"
 
{| class="wikitable" style="text-align:left"
 
!   !!m=2!!m=3
 
!   !!m=2!!m=3
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|-
 
|-
 
|}
 
|}
==== Numbers that have exactly k representations as the sum of m squares >= 0 (List E)====
+
==== Numbers that have exactly k representations as the sum of m squares >= 0 (Table E)====
 
<!--A295158 quant_eq ten representations as the sum of five least_0 pow_2. -->
 
<!--A295158 quant_eq ten representations as the sum of five least_0 pow_2. -->
 
{| class="wikitable" style="text-align:left"
 
{| class="wikitable" style="text-align:left"
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|-
 
|-
 
|}
 
|}
===New Lists===
+
==New Lists==
==== Numbers that can be expressed as the sum of k squares in m or more ways (List R2)====
+
===Squares===
 +
==== Numbers that can be expressed as the sum of k distinct squares in m or more ways (Table R2d)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#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
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| 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=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=2 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of 2 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025313 A025313]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025314 A025314]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025315 A025315]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025316 A025316]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025317 A025317]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025318 A025318]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025319 A025319]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025320 A025320]</span>
 
|-
 
|-
 
| k=3 ||<span title="Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 &lt;= i &lt;= j &lt;= k.">[https://oeis.org/A024796 A024796]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 2 or more ways.">[https://oeis.org/A024804 A024804]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025349 A025349]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025350 A025350]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025351 A025351]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025352 A025352]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025353 A025353]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025354 A025354]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025355 A025355]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025356 A025356]</span>
 
| k=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=4 ||&#xa0;||<span title="Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.">[https://oeis.org/A259058 A259058]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 3 or more ways.">[https://oeis.org/A025387 A025387]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 4 or more ways.">[https://oeis.org/A025388 A025388]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 5 or more ways.">[https://oeis.org/A025389 A025389]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 6 or more ways.">[https://oeis.org/A025390 A025390]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 7 or more ways.">[https://oeis.org/A025391 A025391]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 8 or more ways.">[https://oeis.org/A025392 A025392]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 9 or more ways.">[https://oeis.org/A025393 A025393]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in 10 or more ways.">[https://oeis.org/A025394 A025394]</span>
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k possibly equal squares in m or more ways (Table R2e)====
 +
{| class="wikitable" style="text-align:left"
 +
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3!!m&gt;=4!!m&gt;=5!!m&gt;=6!!m&gt;=7!!m&gt;=8!!m&gt;=9!!m&gt;=10
 +
|-
 +
| k=2 ||&#xa0;||<span title="Numbers that are the sum of 2 nonzero squares in 2 or more ways.">[https://oeis.org/A007692 A007692]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 3 or more ways.">[https://oeis.org/A025294 A025294]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 4 or more ways.">[https://oeis.org/A025295 A025295]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 5 or more ways.">[https://oeis.org/A025296 A025296]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 6 or more ways.">[https://oeis.org/A025297 A025297]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 7 or more ways.">[https://oeis.org/A025298 A025298]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 8 or more ways.">[https://oeis.org/A025299 A025299]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 9 or more ways.">[https://oeis.org/A025300 A025300]</span>||<span title="Numbers that are the sum of 2 nonzero squares in 10 or more ways.">[https://oeis.org/A025301 A025301]</span>
 +
|-
 +
| k=3 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of 3 nonzero squares in 3 or more ways.">[https://oeis.org/A025331 A025331]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 4 or more ways.">[https://oeis.org/A025332 A025332]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 5 or more ways.">[https://oeis.org/A025333 A025333]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 6 or more ways.">[https://oeis.org/A025334 A025334]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 7 or more ways.">[https://oeis.org/A025335 A025335]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 8 or more ways.">[https://oeis.org/A025336 A025336]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 9 or more ways.">[https://oeis.org/A025337 A025337]</span>||<span title="Numbers that are the sum of 3 nonzero squares in 10 or more ways.">[https://oeis.org/A025338 A025338]</span>
 +
|-
 +
| k=4 ||&#xa0;||<span title="Numbers that are the sum of 4 nonzero squares in 2 or more ways.">[https://oeis.org/A025367 A025367]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 3 or more ways.">[https://oeis.org/A025368 A025368]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 4 or more ways.">[https://oeis.org/A025369 A025369]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 5 or more ways.">[https://oeis.org/A025370 A025370]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 6 or more ways.">[https://oeis.org/A025371 A025371]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 7 or more ways.">[https://oeis.org/A025372 A025372]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 8 or more ways.">[https://oeis.org/A025373 A025373]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 9 or more ways.">[https://oeis.org/A025374 A025374]</span>||<span title="Numbers that are the sum of 4 nonzero squares in 10 or more ways.">[https://oeis.org/A025375 A025375]</span>
 
|-
 
|-
 
| k=5 ||&#xa0;||<span title="Numbers that are the sum of five squares in two or more ways.">[https://oeis.org/A344795 A344795]</span>||<span title="Numbers that are the sum of five squares in three or more ways.">[https://oeis.org/A344796 A344796]</span>||<span title="Numbers that are the sum of five squares in four or more ways.">[https://oeis.org/A344797 A344797]</span>||<span title="Numbers that are the sum of five squares in five or more ways.">[https://oeis.org/A344798 A344798]</span>||<span title="Numbers that are the sum of five squares in six or more ways.">[https://oeis.org/A344799 A344799]</span>||<span title="Numbers that are the sum of five squares in seven or more ways.">[https://oeis.org/A344800 A344800]</span>||<span title="Numbers that are the sum of five squares in eight or more ways.">[https://oeis.org/A344801 A344801]</span>||<span title="Numbers that are the sum of five squares in nine or more ways.">[https://oeis.org/A344802 A344802]</span>||<span title="Numbers that are the sum of five squares in ten or more ways.">[https://oeis.org/A344803 A344803]</span>
 
| k=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>
Line 136: Line 148:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k squares in exactly m ways (List S2)====
+
==== Numbers that can be expressed as the sum of k distinct squares in exactly m ways (Table S2d)====
 +
{| class="wikitable" style="text-align:left"
 +
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
 +
|-
 +
| k=2 ||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025302 A025302]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025303 A025303]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025304 A025304]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025305 A025305]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025306 A025306]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025307 A025307]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025308 A025308]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025309 A025309]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025310 A025310]</span>||<span title="Numbers that are the sum of 2 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025311 A025311]</span>
 +
|-
 +
| k=3 ||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly one way.">[https://oeis.org/A025339 A025339]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025340 A025340]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025341 A025341]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025342 A025342]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025343 A025343]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025344 A025344]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025345 A025345]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025346 A025346]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025347 A025347]</span>||<span title="Numbers that are the sum of 3 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025348 A025348]</span>
 +
|-
 +
| k=4 ||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 1 way.">[https://oeis.org/A025376 A025376]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 2 ways.">[https://oeis.org/A025377 A025377]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 3 ways.">[https://oeis.org/A025378 A025378]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 4 ways.">[https://oeis.org/A025379 A025379]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 5 ways.">[https://oeis.org/A025380 A025380]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 6 ways.">[https://oeis.org/A025381 A025381]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 7 ways.">[https://oeis.org/A025382 A025382]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 8 ways.">[https://oeis.org/A025383 A025383]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 9 ways.">[https://oeis.org/A025384 A025384]</span>||<span title="Numbers that are the sum of 4 distinct nonzero squares in exactly 10 ways.">[https://oeis.org/A025385 A025385]</span>
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k possibly equal squares in exactly m ways (Table S2e)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#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=2 ||<span title="Numbers that are the sum of 2 nonzero squares in exactly 1 way.">[https://oeis.org/A025284 A025284]</span>||<span title="Numbers that are the sum of 2 squares in exactly 2 ways.">[https://oeis.org/A085625 A085625]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 3 ways.">[https://oeis.org/A025286 A025286]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 4 ways.">[https://oeis.org/A025287 A025287]</span>||<span title="Numbers that are the sum of 2 squares in exactly 5 ways.">[https://oeis.org/A294716 A294716]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 6 ways.">[https://oeis.org/A025289 A025289]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 7 ways.">[https://oeis.org/A025290 A025290]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 8 ways.">[https://oeis.org/A025291 A025291]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 9 ways.">[https://oeis.org/A025292 A025292]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 10 ways.">[https://oeis.org/A025293 A025293]</span>||<span title="Numbers that are the sum of 2 nonzero squares in exactly 11 ways.">[https://oeis.org/A236711 A236711]</span>
 
|-
 
|-
| k=3 ||<span title="Numbers that are the sum of 3 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=3 ||<span title="Numbers that are the sum of 3 nonzero squares in exactly 1 way.">[https://oeis.org/A025321 A025321]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 2 ways.">[https://oeis.org/A025322 A025322]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 3 ways.">[https://oeis.org/A025323 A025323]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 4 ways.">[https://oeis.org/A025324 A025324]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 5 ways.">[https://oeis.org/A025325 A025325]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 6 ways.">[https://oeis.org/A025326 A025326]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 7 ways.">[https://oeis.org/A025327 A025327]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 8 ways.">[https://oeis.org/A025328 A025328]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 9 ways.">[https://oeis.org/A025329 A025329]</span>||<span title="Numbers that are the sum of 3 nonzero squares in exactly 10 ways.">[https://oeis.org/A025330 A025330]</span>||&#xa0;
 
|-
 
|-
| k=4 ||<span title="Numbers that are the sum of 4 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=4 ||<span title="Numbers that are the sum of 4 nonzero squares in exactly 1 way.">[https://oeis.org/A025357 A025357]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 2 ways.">[https://oeis.org/A025358 A025358]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 3 ways.">[https://oeis.org/A025359 A025359]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 4 ways.">[https://oeis.org/A025360 A025360]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 5 ways.">[https://oeis.org/A025361 A025361]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 6 ways.">[https://oeis.org/A025362 A025362]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 7 ways.">[https://oeis.org/A025363 A025363]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 8 ways.">[https://oeis.org/A025364 A025364]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 9 ways.">[https://oeis.org/A025365 A025365]</span>||<span title="Numbers that are the sum of 4 nonzero squares in exactly 10 ways.">[https://oeis.org/A025366 A025366]</span>||&#xa0;
 
|-
 
|-
 
| k=5 ||<span title="Numbers that are the sum of 5 nonzero squares in exactly 1 way.">[https://oeis.org/A294675 A294675]</span>||<span title="Numbers that have exactly two representations as a sum of five nonnegative squares.">[https://oeis.org/A295150 A295150]</span>||<span title="Numbers that have exactly three representations as a sum of five nonnegative squares.">[https://oeis.org/A295151 A295151]</span>||<span title="Numbers that have exactly four representations as a sum of five nonnegative squares.">[https://oeis.org/A295152 A295152]</span>||<span title="Numbers that have exactly five representations as a sum of five nonnegative squares.">[https://oeis.org/A295153 A295153]</span>||<span title="Numbers that have exactly six representations as a sum of five nonnegative squares.">[https://oeis.org/A295154 A295154]</span>||<span title="Numbers that have exactly seven representations as a sum of five nonnegative squares.">[https://oeis.org/A295155 A295155]</span>||<span title="Numbers that have exactly eight representations as a sum of five nonnegative squares.">[https://oeis.org/A295156 A295156]</span>||<span title="Numbers that have exactly nine representations as a sum of five nonnegative squares.">[https://oeis.org/A295157 A295157]</span>||<span title="Numbers that have exactly ten representations as a sum of five nonnegative squares.">[https://oeis.org/A295158 A295158]</span>||&#xa0;
 
| k=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;
Line 153: Line 176:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k cubes in m or more ways (List R3)====
+
 
 +
=== Cubes ===
 +
==== Numbers that can be expressed as the sum of k distinct cubes in m or more ways (Table R3d)====
 +
{| class="wikitable" style="text-align:left"
 +
! &#xa0; !!m&gt;=1!!m&gt;=2!!m&gt;=3
 +
|-
 +
| k=1 ||<span title="The cubes: a(n) = n^3.">[https://oeis.org/A000578 A000578]</span>||&#xa0;||&#xa0;
 +
|-
 +
| k=2 ||<span title="Taxi-cab numbers: sums of 2 cubes in more than 1 way.">[https://oeis.org/A001235 A001235]</span>||&#xa0;||&#xa0;
 +
|-
 +
| k=3 ||&#xa0;||<span title="Numbers that are the sum of 3 distinct positive cubes in 2 or more ways.">[https://oeis.org/A024974 A024974]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in 3 or more ways.">[https://oeis.org/A025402 A025402]</span>
 +
|-
 +
| k=4 ||&#xa0;||<span title="Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes.">[https://oeis.org/A259060 A259060]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in 3 or more ways.">[https://oeis.org/A025413 A025413]</span>
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k possibly equal cubes in m or more ways (Table R3e)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#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 ||&#xa0;||&#xa0;||<span title="Numbers that are the sum of two positive cubes in at least three ways (all solutions).">[https://oeis.org/A018787 A018787]</span>||<span title="Numbers that are the sum of two positive cubes in at least four ways (all solutions).">[https://oeis.org/A023051 A023051]</span>||<span title="Sum of two positive cubes in at least five ways (all solutions).">[https://oeis.org/A051167 A051167]</span>||&#xa0;||&#xa0;||&#xa0;||&#xa0;||&#xa0;
 
|-
 
|-
| k=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;||&#xa0;||<span title="Numbers that are the sum of 3 positive cubes in 3 or more ways.">[https://oeis.org/A025398 A025398]</span>||<span title="Numbers that are the sum of three positive cubes in four or more ways.">[https://oeis.org/A343968 A343968]</span>||<span title="Numbers that are the sum of three positive cubes in five or more ways.">[https://oeis.org/A343967 A343967]</span>||<span title="Numbers that are the sum of three third powers in six or more ways.">[https://oeis.org/A345083 A345083]</span>||<span title="Numbers that are the sum of three third powers in seven or more ways.">[https://oeis.org/A345086 A345086]</span>||<span title="Numbers that are the sum of three third powers in eight or more ways.">[https://oeis.org/A345087 A345087]</span>||<span title="Numbers that are the sum of three third powers in nine or more ways.">[https://oeis.org/A345119 A345119]</span>||<span title="Numbers that are the sum of three third powers in ten or more ways.">[https://oeis.org/A345121 A345121]</span>
 
|-
 
|-
| k=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 the sum of 4 positive cubes in 2 or more ways.">[https://oeis.org/A025406 A025406]</span>||<span title="Numbers that are the sum of 4 positive cubes in 3 or more ways.">[https://oeis.org/A025407 A025407]</span>||<span title="Numbers that are the sum of four positive cubes in four or more ways.">[https://oeis.org/A343971 A343971]</span>||<span title="Numbers that are the sum of four positive cubes in five or more ways.">[https://oeis.org/A343987 A343987]</span>||<span title="Numbers that are the sum of four third powers in six or more ways.">[https://oeis.org/A345148 A345148]</span>||<span title="Numbers that are the sum of four third powers in seven or more ways.">[https://oeis.org/A345150 A345150]</span>||<span title="Numbers that are the sum of four third powers in eight or more ways.">[https://oeis.org/A345152 A345152]</span>||<span title="Numbers that are the sum of four third powers in nine or more ways.">[https://oeis.org/A345146 A345146]</span>||<span title="Numbers that are the sum of four third powers in ten or more ways.">[https://oeis.org/A345155 A345155]</span>
|-
 
| k=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=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>
Line 178: Line 214:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k cubes in exactly m ways (List S3)====
+
 
 +
==== Numbers that can be expressed as the sum of k distinct cubes in exactly m ways (Table S3d)====
 +
{| class="wikitable" style="text-align:left"
 +
! &#xa0; !!m=1!!m=2!!m=3
 +
|-
 +
| k=3 ||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.">[https://oeis.org/A025399 A025399]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 2 ways.">[https://oeis.org/A025400 A025400]</span>||<span title="Numbers that are the sum of 3 distinct positive cubes in exactly 3 ways.">[https://oeis.org/A025401 A025401]</span>
 +
|-
 +
| k=4 ||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 1 way.">[https://oeis.org/A025408 A025408]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 2 ways.">[https://oeis.org/A025409 A025409]</span>||<span title="Numbers that are the sum of 4 distinct positive cubes in exactly 3 ways.">[https://oeis.org/A025410 A025410]</span>
 +
|-
 +
|}
 +
==== Numbers that can be expressed as the sum of k possibly equal cubes in exactly m ways (Table S3e)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
Line 184: Line 230:
 
| 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=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=3 ||<span title="Numbers that are the sum of 3 positive cubes in exactly 1 way.">[https://oeis.org/A025395 A025395]</span>||<span title="Numbers that are the sum of 3 positive cubes in exactly 2 ways.">[https://oeis.org/A025396 A025396]</span>||<span title="Numbers that are the sum of 3 positive cubes in exactly 3 ways.">[https://oeis.org/A025397 A025397]</span>||<span title="Numbers that are the sum of three positive cubes in exactly 4 ways.">[https://oeis.org/A343969 A343969]</span>||<span title="Numbers that are the sum of three positive cubes in exactly five ways.">[https://oeis.org/A343970 A343970]</span>||<span title="Numbers that are the sum of three third powers in exactly six ways.">[https://oeis.org/A345084 A345084]</span>||<span title="Numbers that are the sum of three third powers in exactly seven ways.">[https://oeis.org/A345085 A345085]</span>||<span title="Numbers that are the sum of three third powers in exactly eight ways.">[https://oeis.org/A345088 A345088]</span>||<span title="Numbers that are the sum of three third powers in exactly nine ways.">[https://oeis.org/A345120 A345120]</span>||<span title="Numbers that are the sum of three third powers in exactly ten ways.">[https://oeis.org/A345122 A345122]</span>
 
|-
 
|-
| k=4 ||<span title="Numbers that are the sum of 4 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=4 ||<span title="Numbers that are the sum of 4 positive cubes in exactly 1 way.">[https://oeis.org/A025403 A025403]</span>||<span title="Numbers that are the sum of 4 positive cubes in exactly 2 ways.">[https://oeis.org/A025404 A025404]</span>||<span title="Numbers that are the sum of 4 positive cubes in exactly 3 ways.">[https://oeis.org/A025405 A025405]</span>||<span title="Numbers that are the sum of four positive cubes in exactly four ways.">[https://oeis.org/A343972 A343972]</span>||<span title="Numbers that are the sum of four positive cubes in exactly five ways.">[https://oeis.org/A343988 A343988]</span>||<span title="Numbers that are the sum of four third powers in exactly six ways.">[https://oeis.org/A345149 A345149]</span>||<span title="Numbers that are the sum of four third powers in exactly seven ways.">[https://oeis.org/A345151 A345151]</span>||<span title="Numbers that are the sum of four third powers in exactly eight ways.">[https://oeis.org/A345153 A345153]</span>||<span title="Numbers that are the sum of four third powers in exactly nine ways.">[https://oeis.org/A345154 A345154]</span>||<span title="Numbers that are the sum of four third powers in exactly ten ways.">[https://oeis.org/A345156 A345156]</span>
 
|-
 
|-
 
| k=5 ||<span title="Numbers that are the sum of 5 positive cubes in exactly 1 way.">[https://oeis.org/A048926 A048926]</span>||<span title="Numbers that are the sum of 5 positive cubes in exactly 2 ways.">[https://oeis.org/A048927 A048927]</span>||<span title="Numbers that are the sum of five positive cubes in exactly three ways.">[https://oeis.org/A343705 A343705]</span>||<span title="Numbers that are the sum of five positive cubes in exactly four ways.">[https://oeis.org/A344035 A344035]</span>||&#xa0;||<span title="Numbers that are the sum of five third powers in exactly six ways.">[https://oeis.org/A345175 A345175]</span>||<span title="Numbers that are the sum of five third powers in exactly seven ways.">[https://oeis.org/A345181 A345181]</span>||<span title="Numbers that are the sum of five third powers in exactly eight ways.">[https://oeis.org/A345184 A345184]</span>||<span title="Numbers that are the sum of five third powers in exactly nine ways.">[https://oeis.org/A345186 A345186]</span>||<span title="Numbers that are the sum of five third powers in exactly ten ways.">[https://oeis.org/A345188 A345188]</span>
 
| k=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>
Line 201: Line 247:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k fourth powers in m or more ways (List R4)====
+
 
 +
=== 4th powers ===
 +
==== Numbers that can be expressed as the sum of k fourth powers in m or more ways (Table R4)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#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
Line 224: Line 272:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k fourth powers in exactly m ways (List S4)====
+
==== Numbers that can be expressed as the sum of k fourth powers in exactly m ways (Table S4)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10
Line 247: Line 295:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k fifth powers in m or more ways (List R5)====
+
 
 +
=== 5th powers ===
 +
==== Numbers that can be expressed as the sum of k fifth powers in m or more ways (Table R5)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#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
Line 270: Line 320:
 
|-
 
|-
 
|}
 
|}
==== Numbers that can be expressed as the sum of k fifth powers in exactly m ways (List S5)====
+
==== Numbers that can be expressed as the sum of k fifth powers in exactly m ways (Table S5)====
 
{| class="wikitable" style="text-align:left"
 
{| 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
 
! &#xa0; !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10

Latest revision as of 07:29, 20 August 2021

Contents

Sums of like powers

Sums of k m-th powers >= 0 (Table 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 (Table 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 (Table 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 (Table 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 (Table 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

Squares

Numbers that can be expressed as the sum of k distinct squares in m or more ways (Table R2d)

  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     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

Numbers that can be expressed as the sum of k possibly equal squares in m or more ways (Table R2e)

  m>=1 m>=2 m>=3 m>=4 m>=5 m>=6 m>=7 m>=8 m>=9 m>=10
k=2   A007692 A025294 A025295 A025296 A025297 A025298 A025299 A025300 A025301
k=3     A025331 A025332 A025333 A025334 A025335 A025336 A025337 A025338
k=4   A025367 A025368 A025369 A025370 A025371 A025372 A025373 A025374 A025375
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 distinct squares in exactly m ways (Table S2d)

  m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10
k=2 A025302 A025303 A025304 A025305 A025306 A025307 A025308 A025309 A025310 A025311
k=3 A025339 A025340 A025341 A025342 A025343 A025344 A025345 A025346 A025347 A025348
k=4 A025376 A025377 A025378 A025379 A025380 A025381 A025382 A025383 A025384 A025385

Numbers that can be expressed as the sum of k possibly equal squares in exactly m ways (Table S2e)

  m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11
k=2 A025284 A085625 A025286 A025287 A294716 A025289 A025290 A025291 A025292 A025293 A236711
k=3 A025321 A025322 A025323 A025324 A025325 A025326 A025327 A025328 A025329 A025330  
k=4 A025357 A025358 A025359 A025360 A025361 A025362 A025363 A025364 A025365 A025366  
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  

Cubes

Numbers that can be expressed as the sum of k distinct cubes in m or more ways (Table R3d)

  m>=1 m>=2 m>=3
k=1 A000578    
k=2 A001235    
k=3   A024974 A025402
k=4   A259060 A025413

Numbers that can be expressed as the sum of k possibly equal cubes in m or more ways (Table R3e)

  m>=1 m>=2 m>=3 m>=4 m>=5 m>=6 m>=7 m>=8 m>=9 m>=10
k=2     A018787 A023051 A051167          
k=3     A025398 A343968 A343967 A345083 A345086 A345087 A345119 A345121
k=4 A003327 A025406 A025407 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 distinct cubes in exactly m ways (Table S3d)

  m=1 m=2 m=3
k=3 A025399 A025400 A025401
k=4 A025408 A025409 A025410

Numbers that can be expressed as the sum of k possibly equal cubes in exactly m ways (Table S3e)

  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 A025395 A025396 A025397 A343969 A343970 A345084 A345085 A345088 A345120 A345122
k=4 A025403 A025404 A025405 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

4th powers

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

5th powers

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