# Difference between revisions of "Talk:Main Page"

From teherba.org

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=== Sums of like powers=== | === Sums of like powers=== | ||

− | ==== Sums of k m-th powers >= 0 ( | + | ==== 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 ( | + | ==== 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 ( | + | ==== 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 ( | + | ==== 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 ( | + | ==== 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" | ||

Line 110: | Line 110: | ||

|- | |- | ||

|} | |} | ||

− | + | ==New Lists== | |

− | ==== Numbers that can be expressed as the sum of k squares in m or more ways ( | + | ===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" | ||

!   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

Line 117: | Line 118: | ||

| k=1 ||<span title="The squares: a(n) = n^2.">[https://oeis.org/A000290 A000290]</span>|| || || || || || || || ||  | | k=1 ||<span title="The squares: a(n) = n^2.">[https://oeis.org/A000290 A000290]</span>|| || || || || || || || ||  | ||

|- | |- | ||

− | | k=2 || || | + | | k=2 || || ||<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 <= i <= j <= 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 <= i <= j <= 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 || ||<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 || ||<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" | ||

+ | !   !!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 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 || || ||<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 || ||<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 || ||<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 || ||<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> | ||

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|- | |- | ||

|} | |} | ||

− | ==== Numbers that can be expressed as the sum of k squares in exactly m ways ( | + | ==== Numbers that can be expressed as the sum of k distinct squares in exactly m ways (Table S2d)==== |

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!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" | ||

!   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10!!m=11 | !   !!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 | + | | 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 | + | | 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>||  |

|- | |- | ||

− | | k=4 ||<span title="Numbers that are the sum of 4 | + | | 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>||  |

|- | |- | ||

| 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>||  | | 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>||  | ||

Line 153: | Line 176: | ||

|- | |- | ||

|} | |} | ||

− | ==== Numbers that can be expressed as the sum of k cubes in m or more ways ( | + | |

+ | === 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" | ||

+ | !   !!m>=1!!m>=2!!m>=3 | ||

+ | |- | ||

+ | | k=1 ||<span title="The cubes: a(n) = n^3.">[https://oeis.org/A000578 A000578]</span>|| ||  | ||

+ | |- | ||

+ | | k=2 ||<span title="Taxi-cab numbers: sums of 2 cubes in more than 1 way.">[https://oeis.org/A001235 A001235]</span>|| ||  | ||

+ | |- | ||

+ | | k=3 || ||<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 || ||<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" | ||

!   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

|- | |- | ||

− | | k= | + | | k=2 || || ||<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>|| || || || ||  |

|- | |- | ||

− | | k= | + | | k=3 || || ||<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= | + | | k=4 ||<span title="Numbers that are the sum of 4 positive cubes in 1 or more way.">[https://oeis.org/A003327 A003327]</span>||<span title="Numbers that are the sum of 4 positive cubes in 2 or more ways.">[https://oeis.org/A025406 A025406]</span>||<span title="Numbers that are the sum of 4 positive cubes in 3 or more ways.">[https://oeis.org/A025407 A025407]</span>||<span title="Numbers that are the sum of four positive cubes in four or more ways.">[https://oeis.org/A343971 A343971]</span>||<span title="Numbers that are the sum of four positive cubes in five or more ways.">[https://oeis.org/A343987 A343987]</span>||<span title="Numbers that are the sum of four third powers in six or more ways.">[https://oeis.org/A345148 A345148]</span>||<span title="Numbers that are the sum of four third powers in seven or more ways.">[https://oeis.org/A345150 A345150]</span>||<span title="Numbers that are the sum of four third powers in eight or more ways.">[https://oeis.org/A345152 A345152]</span>||<span title="Numbers that are the sum of four third powers in nine or more ways.">[https://oeis.org/A345146 A345146]</span>||<span title="Numbers that are the sum of four third powers in ten or more ways.">[https://oeis.org/A345155 A345155]</span> |

− | |||

− | |||

|- | |- | ||

| k=5 || ||<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 || ||<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 ( | + | |

+ | ==== Numbers that can be expressed as the sum of k distinct cubes in exactly m ways (Table S3d)==== | ||

+ | {| class="wikitable" style="text-align:left" | ||

+ | !   !!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" | ||

!   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 | !   !!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>|| || || || || ||  | | 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>|| || || || || ||  | ||

|- | |- | ||

− | | k=3 ||<span title="Numbers that are the sum of 3 | + | | 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 | + | | 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>|| ||<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>|| ||<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 ( | + | |

+ | === 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" | ||

!   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

Line 224: | Line 272: | ||

|- | |- | ||

|} | |} | ||

− | ==== Numbers that can be expressed as the sum of k fourth powers in exactly m ways ( | + | ==== 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" | ||

!   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 | !   !!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 ( | + | |

+ | === 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" | ||

!   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | !   !!m>=1!!m>=2!!m>=3!!m>=4!!m>=5!!m>=6!!m>=7!!m>=8!!m>=9!!m>=10 | ||

Line 270: | Line 320: | ||

|- | |- | ||

|} | |} | ||

− | ==== Numbers that can be expressed as the sum of k fifth powers in exactly m ways ( | + | ==== 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" | ||

!   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 | !   !!m=1!!m=2!!m=3!!m=4!!m=5!!m=6!!m=7!!m=8!!m=9!!m=10 |

## Revision as of 08:29, 20 August 2021

## Contents

- 1 Sums of like powers
- 2 New Lists
- 2.1 Squares
- 2.1.1 Numbers that can be expressed as the sum of k distinct squares in m or more ways (Table R2d)
- 2.1.2 Numbers that can be expressed as the sum of k possibly equal squares in m or more ways (Table R2e)
- 2.1.3 Numbers that can be expressed as the sum of k distinct squares in exactly m ways (Table S2d)
- 2.1.4 Numbers that can be expressed as the sum of k possibly equal squares in exactly m ways (Table S2e)

- 2.2 Cubes
- 2.2.1 Numbers that can be expressed as the sum of k distinct cubes in m or more ways (Table R3d)
- 2.2.2 Numbers that can be expressed as the sum of k possibly equal cubes in m or more ways (Table R3e)
- 2.2.3 Numbers that can be expressed as the sum of k distinct cubes in exactly m ways (Table S3d)
- 2.2.4 Numbers that can be expressed as the sum of k possibly equal cubes in exactly m ways (Table S3e)

- 2.3 4th powers
- 2.4 5th powers

- 2.1 Squares

### 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 |