A003325 -id:A003325 - cp's OEIS frontend

A003333 Numbers that are the sum of 10 positive cubes.

10, 17, 24, 31, 36, 38, 43, 45, 50, 52, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 108, 109, 111, 113, 114, 115, 116, 118, 120, 121, 122, 123, 125, 127, 128, 129, 130, 132, 134, 135, 136, 137, 139, 140, 141, 142

Offset: 1

N. J. A. Sloane

374 is the largest of only 99 positive integers not in this sequence. - M. F. Hasler, Aug 13 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1646 is in the sequence as 1646 = 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 7^3 + 7^3 +  8^3.
2790 is in the sequence as 2790 = 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 6^3 + 6^3 + 7^3 + 8^3 + 10^3.
3450 is in the sequence as 3450 = 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 7^3 + 8^3 + 8^3 + 9^3 +  9^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
OEIS Wiki, Index to sequences related to sums of like powers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Other sequences of numbers that are the sum of x nonzero y-th powers:
  A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
  A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
  A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
  A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
  A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
  A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

(A003333_upto(N)=select( {is_A003333(n,k=10,m=3,L=sqrtnint(abs(n-k+1),m))=if( n>k*L^m || nM. F. Hasler, Aug 02 2020
A3333=A003333_upto(320); A003333(n)=if(n>275, n+99, n>222, n+98, A3333[n]) \\ M. F. Hasler, Aug 13 2020

a(n) = n + 99 for all n > 275. - M. F. Hasler, Aug 13 2020

A003369 Numbers that are the sum of 2 positive 7th powers.

Original entry on oeis.org

2, 129, 256, 2188, 2315, 4374, 16385, 16512, 18571, 32768, 78126, 78253, 80312, 94509, 156250, 279937, 280064, 282123, 296320, 358061, 559872, 823544, 823671, 825730, 839927, 901668, 1103479, 1647086, 2097153, 2097280, 2099339, 2113536

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
3909794986386 is in the sequence as 3909794986386 = 57^7 + 57^7.
6061605477062 is in the sequence as 6061605477062 = 19^7 + 67^7.
26019535290982 is in the sequence as 26019535290982 = 61^7 + 81^7. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000404 (2 squares), A003325 (2 cubes), A003336 (2 4th), A003347 (2 5th), A003358 (2 6th), A088719 (2 distinct 7th), A003380 (2 8th).

Cf. A001015 (seventh powers).

N:= 10^7: # to get all terms <= N
S:= select(`<=`, {seq(seq(a^7+b^7, a=1..b), b=1..floor(N^(1/7)))}, N):
sort(convert(S, list)); # Robert Israel, Sep 03 2017

lst={}; Do[If[(a^7+b^7)==n, AppendTo[lst, n]], {n, 200000}, {a, (n/2)^(1/7)}, {b, a, (n-a^7)^(1/7)}]; lst (* XU Pingya, Sep 03 2017 *)
Module[{upto=10},Select[Union[Total/@Tuples[Range[upto]^7,2]],#<= (upto^7)&]] (* Harvey P. Dale, Feb 04 2019 *)

A024670 Numbers that are sums of 2 distinct positive cubes.

Original entry on oeis.org

9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 513, 520, 539, 559, 576, 637, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1512, 1547, 1674

Offset: 1

Clark Kimberling

nonn

This sequence contains no primes since x^3+y^3=(x^2-x*y+y^2)*(x+y). - M. F. Hasler, Apr 12 2008
There are no terms == 3, 4, 5 or 6 mod 9. - Robert Israel, Oct 07 2014
a(n) mod 2: {1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,1,0, ...} - Daniel Forgues, Sep 27 2018

			9 is in the sequence since 2^3 + 1^3 = 9.
35 is in the sequence since 3^3 + 2^3 = 35.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..902 from M. F. Hasler)
Index to sequences related to sums of cubes

See also: Sums of 2 positive cubes (not necessarily distinct): A003325. Sums of 3 distinct positive cubes: A024975. Sums of distinct positive cubes: A003997. Sums of 2 distinct nonnegative cubes: A114090. Sums of 2 nonnegative cubes: A004999. Sums of 2 distinct positive squares: A004431. Cubes: A000578.
Cf. A373971 (characteristic function).
Indices of nonzero terms in A025468.

N:= 10000: # to get all terms <= N
S:= select(`<=`,{seq(seq(i^3 + j^3, j = 1 .. i-1), i = 2 .. floor(N^(1/3)))},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(S,list));
# Robert Israel, Oct 07 2014

lst={};Do[Do[x=a^3;Do[y=b^3;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/3)],a+1,-1}],{a,Floor[n^(1/3)],1,-1}],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Select[Range@ 1700, Total@ Boole@ Map[And[! MemberQ[#, 0], UnsameQ @@ #] &, PowersRepresentations[#, 2, 3]] > 0 &] (* Michael De Vlieger, May 13 2017 *)

isA024670(n)=for( i=ceil(sqrtn( n\2+1,3)),sqrtn(n-.5,3), isA000578(n-i^3) & return(1)) /* One could also use "for( i=2,sqrtn( n\2-1,3),...)" but this is much slower since there are less cubes in [n/2,n] than in [1,n/2]. Replacing the -1 here by +.5 would yield A003325, allowing for a(n)=x^3+x^3. Replacing -1 by 0 may miss some a(n) of this form due to rounding errors. - M. F. Hasler, Apr 12 2008 */

from itertools import count, takewhile
def aupto(limit):
    cbs = list(takewhile(lambda x: x <= limit, (i**3 for i in count(1))))
    sms = set(c+d for i, c in enumerate(cbs) for d in cbs[i+1:])
    return sorted(s for s in sms if s <= limit)
print(aupto(1674)) # Michael S. Branicky, Sep 28 2021

Name edited by Zak Seidov, May 31 2011

A003390 Sum of 12 nonzero 8th powers.

Original entry on oeis.org

12, 267, 522, 777, 1032, 1287, 1542, 1797, 2052, 2307, 2562, 2817, 3072, 6572, 6827, 7082, 7337, 7592, 7847, 8102, 8357, 8612, 8867, 9122, 9377, 13132, 13387, 13642, 13897, 14152, 14407, 14662, 14917, 15172, 15427, 15682, 19692, 19947, 20202, 20457

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1890948 is in the sequence as 1890948 = 2^8 + 2^8 + 2^8 + 4^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 5^8 + 5^8 + 5^8.
2338951 is in the sequence as 2338951 = 1^8 + 1^8 + 1^8 + 1^8 + 1^8 + 3^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 6^8.
3841896 is in the sequence as 3841896 = 1^8 + 1^8 + 1^8 + 2^8 + 3^8 + 3^8 + 3^8 + 3^8 + 4^8 + 5^8 + 6^8 + 6^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (replacing a b-file that missed terms)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2), A003072 (3, 3), A003325 (2, 3), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003386 (8, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11).

A003390_upto(N=1e5, n=12, p=8)={my(P=[x^p|x<-[1..sqrtnint(N-n+1, p)]], S=P); while(n--, S=Set(concat([[x+y|y<-S, x+y<=N]|x<-P]))); S} \\ M. F. Hasler, Jul 03 2025

Removed incorrect program, offset corrected by David A. Corneth, Aug 01 2020

A004802 Numbers that are the sum of 2 nonzero 10th powers.

Original entry on oeis.org

2, 1025, 2048, 59050, 60073, 118098, 1048577, 1049600, 1107625, 2097152, 9765626, 9766649, 9824674, 10814201, 19531250, 60466177, 60467200, 60525225, 61514752, 70231801, 120932352, 282475250, 282476273, 282534298, 283523825, 292240874

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1103972715709403850 is in the sequence as 1103972715709403850 = 51^10 + 63^10.
2059617246125773226 is in the sequence as 2059617246125773226 = 61^10 + 65^10.
27850192968371852849 is in the sequence as 27850192968371852849 = 25^10 + 88^10. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Removed incorrect program. - David A. Corneth, Aug 01 2020

A003338 Numbers that are the sum of 4 nonzero 4th powers.

Original entry on oeis.org

4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153, 1218

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4.
81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4.
106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

# returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d.
A003338 := proc(n)
    local a,i,j,k,l,res ;
    a := 0 ;
    for i from 1 do
        if i^4 > n then
            break ;
        end if;
        for j from i do
            if i^4+j^4 > n then
                break ;
            end if;
            for k from j do
                if i^4+j^4+k^4> n then
                    break;
                end if;
                res := n-i^4-j^4-k^4 ;
                if issqr(res) then
                    res := sqrt(res) ;
                    if issqr(res) then
                        l := sqrt(res) ;
                        if l >= k then
                            a := a+1 ;
                        end if;
                    end if;
                end if;
            end do:
        end do:
    end do:
    a ;
end proc:
for n from 1 do
    if A003338(n) > 0 then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 17 2023

f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]],seq}, seq=Union[Total/@(Tuples[Range[nn],{4}]^4)]; Select[seq,#<=maxno&]]
f[1000] (* Harvey P. Dale, Feb 27 2011 *)

limit = 1218
from functools import lru_cache
qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
qds = set(qd)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n, )} if n in qds else set()
  return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # Michael S. Branicky, Apr 19 2021

A003330 Numbers that are the sum of 7 positive cubes.

Original entry on oeis.org

7, 14, 21, 28, 33, 35, 40, 42, 47, 49, 54, 56, 59, 61, 66, 68, 70, 73, 75, 77, 80, 84, 85, 87, 91, 92, 94, 96, 98, 99, 103, 105, 106, 110, 111, 112, 113, 117, 118, 122, 124, 125, 129, 131, 132, 133, 136, 137, 138, 140, 143, 144, 145, 147, 148, 150, 151, 152, 154

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020

2408 is the largest among only 208 positive integers not in this sequence: cf. formula. - M. F. Hasler, Aug 23 2020

			From _M. F. Hasler_, Aug 23 2020: (Start)
The first few terms are multiples of 7 because of the coincidence that 2^3 - 1^3 = 7, equal to the number of cubes we consider here:
7 = 1^3 * 7 is the smallest sum of seven positive cubes.
14 = 1^3 * 6 + 2^3 = 6 + 8 is the next larger sum of seven positive cubes.
21 = 1^3 * 5 + 2^3 * 2 = 5 + 16 is the next larger sum of seven positive cubes.
28 = 1^3 * 4 + 2^3 * 3 = 4 + 24 is the next larger sum of seven positive cubes.
There are three more terms of this form, but the next larger sum of seven positive cubes is a(5) = 3^3 + 6 * 1^3 = 33. (End)
From _David A. Corneth_, Aug 01 2020: (Start)
2070 is in the sequence as 2070 = 4^3 + 4^3 + 4^3 + 5^3 + 8^3 + 8^3 +  9^3.
2383 is in the sequence as 2383 = 3^3 + 5^3 + 5^3 + 6^3 + 6^3 + 7^3 + 11^3.
3592 is in the sequence as 3592 = 4^3 + 5^3 + 6^3 + 9^3 + 9^3 + 9^3 + 10^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000
OEIS Wiki, Index to sequences related to sums of like powers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences related to sums of cubes

Cf. A000578, A004829, A003331, A003341.
Other sequences of numbers that are the sum of x nonzero y-th powers:
  A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
  A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
  A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
  A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
  A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
  A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

(A003330_upto(N, k=7, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(160) \\ M. F. Hasler, Aug 02 2020

a(n) = n + 208 for all n > 2200. - M. F. Hasler, Aug 23 2020

More terms from Arlin Anderson (starship1(AT)gmail.com)

A003331 Numbers that are the sum of 8 positive cubes.

Original entry on oeis.org

8, 15, 22, 29, 34, 36, 41, 43, 48, 50, 55, 57, 60, 62, 64, 67, 69, 71, 74, 76, 78, 81, 83, 85, 86, 88, 92, 93, 95, 97, 99, 100, 102, 104, 106, 107, 111, 112, 113, 114, 118, 119, 120, 121, 123, 125, 126, 130, 132, 133, 134, 137, 138, 139, 140, 141, 144, 145, 146, 148, 149

Offset: 1

N. J. A. Sloane

620 is the largest among only 142 positive integers not in this sequence. This can be proved by induction. - M. F. Hasler, Aug 13 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1796 is in the sequence as 1796 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 7^3 + 7^3 + 9^3.
2246 is in the sequence as 2246 = 2^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 7^3 + 11^3.
3164 is in the sequence as 3164 = 5^3 + 5^3 + 6^3 + 6^3 + 8^3 + 8^3 + 9^3 + 9^3.(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
OEIS Wiki, Index to sequences related to sums of like powers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Other sequences of numbers that are the sum of x nonzero y-th powers:
  A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
  A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
  A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
  A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
  A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
  A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

Module[{upto=200,c},c=Floor[Surd[upto,3]];Select[Union[Total/@ Tuples[ Range[ c]^3,8]],#<=upto&]] (* Harvey P. Dale, Jan 11 2016 *)

(A003331_upto(N, k=8, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ M. F. Hasler, Aug 02 2020
A003331(n)=if(n>478, n+142, n>329, n+141, A003331_upto(470)[n]) \\ M. F. Hasler, Aug 13 2020

from itertools import combinations_with_replacement as mc
def aupto(lim):
    cbs = (i**3 for i in range(1, int((lim-7)**(1/3))+2))
    return sorted(set(k for k in (sum(c) for c in mc(cbs, 8)) if k <= lim))
print(aupto(150)) # Michael S. Branicky, Aug 15 2021

a(n) = 142 + n for all n > 478. - M. F. Hasler, Aug 13 2020

A003356 Numbers that are the sum of 11 positive 5th powers.

Original entry on oeis.org

11, 42, 73, 104, 135, 166, 197, 228, 253, 259, 284, 290, 315, 321, 346, 352, 377, 408, 439, 470, 495, 501, 526, 532, 557, 563, 588, 619, 650, 681, 712, 737, 743, 768, 774, 799, 830, 861, 892, 923, 954, 979, 985, 1010, 1034, 1041, 1065, 1072, 1096, 1103, 1127, 1134

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
16989 is in the sequence as 16989 = 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 4^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5.
22564 is in the sequence as 22564 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 4^5 + 4^5 + 5^5 + 7^5.
30191 is in the sequence as 30191 = 1^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 4^5 + 5^5 + 6^5 + 7^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000584 (fifth powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Incorrect program removed by David A. Corneth, Aug 01 2020

A003332 Numbers that are the sum of 9 positive cubes.

Original entry on oeis.org

9, 16, 23, 30, 35, 37, 42, 44, 49, 51, 56, 58, 61, 63, 65, 68, 70, 72, 75, 77, 79, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 107, 108, 110, 112, 113, 114, 115, 119, 120, 121, 122, 124, 126, 127, 128, 129, 131, 133, 134, 135, 138, 139, 140, 141, 142, 145, 146, 147

Offset: 1

N. J. A. Sloane

422 and 471 are the two largest of only 114 positive integers not in this sequence. This can be proved by induction. - M. F. Hasler, Aug 13 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
1352 is in the sequence as 1352 = 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 6^3 + 6^3 + 8^3.
2312 is in the sequence as 2312 = 5^3 + 5^3 + 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 7^3 + 8^3.
3383 is in the sequence as 3383 = 4^3 + 5^3 + 5^3 + 5^3 + 6^3 + 6^3 + 8^3 + 10^3 + 10^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
OEIS Wiki, Index to sequences related to sums of like powers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. numbers that are the sum of x nonzero y-th powers:
  A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
  A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
  A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
  A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
  A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
  A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

With[{upto=150},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-8,3]]]^3, 9]],#<=upto&]](* Harvey P. Dale, Jan 04 2015 *)

(A003332_upto(N, k=9, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(160) \\ See also A003333 for alternate code. - M. F. Hasler, Aug 02 2020
A003332(n)=if(n>357, n+114, A003332_upto(471)[n]) \\ M. F. Hasler, Aug 13 2020

a(n) = 114 + n for all n > 357. - M. F. Hasler, Aug 13 2020

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A003333 Numbers that are the sum of 10 positive cubes.

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A003369 Numbers that are the sum of 2 positive 7th powers.

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A024670 Numbers that are sums of 2 distinct positive cubes.

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A003390 Sum of 12 nonzero 8th powers.

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A004802 Numbers that are the sum of 2 nonzero 10th powers.

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A003338 Numbers that are the sum of 4 nonzero 4th powers.

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A003330 Numbers that are the sum of 7 positive cubes.

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