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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