A003328 -id:A003328 - cp's OEIS frontend

A004814 Numbers that are the sum of 3 positive 11th powers.

3, 2050, 4097, 6144, 177149, 179196, 181243, 354295, 356342, 531441, 4194306, 4196353, 4198400, 4371452, 4373499, 4548598, 8388609, 8390656, 8565755, 12582912, 48828127, 48830174, 48832221, 49005273, 49007320, 49182419, 53022430

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)
204800049005272 is in the sequence as 204800049005272 = 3^11 + 5^11 + 20^11.
2518268235958260 is in the sequence as 2518268235958260 = 16^11 + 19^11 + 25^11.
3786934745885995 is in the sequence as 3786934745885995 = 10^11 + 19^11 + 26^11. (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

A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710

Offset: 1

Eric W. Weisstein

nonn

It appears that this sequence has 15416 terms, the last of which is 2243453. - Donovan Johnson, Jan 11 2013

From a(1) = 157 we see that c(n) = (number of ways n is the sum of 5 cubes) coincides with A010057 = characteristic function of cubes, up to n = 156. This sequence lists the numbers n for which c(n) = 2. See A003328 for c(n) > 0 and A048926 for c(n) = 1. - M. F. Hasler, Jan 04 2023

Donovan Johnson, Table of n, a(n) for n = 1..15416 (terms < 10^8)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003328 (sums of 5 positive cubes), A025404, A048926 (sum of 5 positive cubes in exactly 1 way), A048930, A294736, A343702, A343705, A344237.

Select[ Range[ 1000], (test = Length[ Select[ PowersRepresentations[#, 5, 3], And @@ (Positive /@ #)& ] ] == 2; If[test, Print[#]]; test)& ](* Jean-François Alcover, Nov 09 2012 *)

(waycount(n,numcubes,imax)={if(numcubes==0, !n, sum(i=1,imax, waycount(n-i^3,numcubes-1,i)))}); isA048927(n)=(waycount(n,5,floor(n^(1/3)))==2); \\ Michael B. Porter, Sep 27 2009

def ways (n, left = 5, last = 1):
  a = last; a3 = a**3; c = 0
  while a3 <= n-left+1:
    if left > 1:
       c += ways(n-a3, left-1, a)
    elif a3 == n:
       c += 1
    a += 1; a3 = a**3
  return c
for n in range (1,1000): # to print this sequence
  if ways(n)==2: print(n,end=", ") # in Python2 use, e.g.: print n,
# Minor edits by M. F. Hasler, Jan 04 2023

More terms from Walter Hofmann (walterh(AT)gmx.de), Jun 01 2000

A048926 Numbers that are the sum of 5 positive cubes in exactly 1 way.

Original entry on oeis.org

5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 64, 68, 71, 75, 78, 82, 83, 89, 90, 94, 96, 97, 101, 108, 109, 115, 116, 120, 127, 129, 131, 134, 135, 136, 138, 143, 145, 146, 150, 152, 153, 155, 162, 164, 169, 171, 172, 176, 181, 183, 188, 190, 192, 194, 195

Offset: 1

Eric W. Weisstein

It appears that this sequence has 11062 terms, the last of which is 1685758. This means that all numbers greater than 1685758 can be written as the sum of five positive cubes in at least two ways. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..11062
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003328, A048927.

Cf. A057906 (numbers not the sum of five positive cubes)

Select[ Range[200], Count[ PowersRepresentations[#, 5, 3], r_ /; FreeQ[r, 0]] == 1 &] (* Jean-François Alcover, Oct 23 2012 *)

from collections import Counter
from itertools import combinations_with_replacement as combs_with_rep
def aupto(lim):
  s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
  s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_with_rep(s, 5)))
  s2counts = Counter(s2)
  return sorted(k for k in s2counts if s2counts[k] == 1)
print(aupto(196)) # Michael S. Branicky, May 12 2021

A343705 Numbers that are the sum of five positive cubes in exactly three ways.

Original entry on oeis.org

766, 810, 827, 829, 865, 883, 981, 1018, 1025, 1044, 1070, 1105, 1108, 1142, 1145, 1161, 1168, 1226, 1233, 1259, 1289, 1350, 1368, 1424, 1431, 1439, 1441, 1457, 1487, 1492, 1494, 1529, 1531, 1538, 1550, 1555, 1568, 1583, 1587, 1592, 1593, 1594, 1609, 1611, 1613, 1639, 1648, 1665, 1672, 1674, 1688, 1707, 1711

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A343704 at term 20 because 1252 = 1^3 + 1^3 + 5^3 + 5^3 + 10^3 = 1^3 + 2^3 + 3^3 + 6^3 + 10^3 = 3^3 + 3^3 + 7^3 + 7^3 + 8^3 = 3^3 + 4^3 + 6^3 + 6^3 + 9^3. Thus this term is in A343704 but not in this sequence.
Comment from D. S. McNeil, May 13 2021: (Start)
If we weaken positive cubes to nonnegative cubes, Deshouillers, Hennecart, and Landreau (2000) give numerical and heuristic evidence that all numbers past 7373170279850 are representable as the sum of 4 nonnegative cubes.
So if they are right, then eventually we can just take some N and represent each of (N-1^3, N-2^3, N-3^3, N-4^3) as the sum of four cubes and then take 1^3, 2^3, 3^3, or 4^3 as our fifth cube, giving at least four 5-cube representations for N.
So it is very likely that the set of numbers representable by the sum of 5 positive cubes in exactly three ways is finite. (End)
It is conjectured that the number of ways of writing N as a sum of 5 positive cubes grows like C(N)*N^(2/3), where C(N) depends on N but is bounded away from zero by an absolute constant (Vaughan, 1981; Vaughan and Wooley, 2002). So the number will exceed 3 as soon as N is large enough, and so it is very likely that this sequence is finite. However, at present this is an open question. - N. J. A. Sloane, May 15 2021 (based on correspondence with Robert Vaughan and Trevor Wooley).

			827 is a term of this sequence because 827 = 1^3 + 4^3 + 5^3 + 5^3 + 8^3 = 2^3 + 2^3 + 5^3 + 7^3 + 7^3 = 2^3 + 3^3 + 4^3 + 6^3 + 8^3.

R. C. Vaughan, The Hardy-Littlewood Method, Cambridge University Press, 1981.
R. C. Vaughan, Trevor Wooley (2002), Waring's Problem: A Survey. In Michael A. Bennet, Bruce C. Berndt, Nigel Boston, Harold G. Diamond, Adolf J. Hildebrand, Walter Philipp (eds.). Number Theory for the Millennium. III. Natick, MA: A. K. Peters, pp. 301-340.

David Consiglio, Jr. and Sean A. Irvine, Table of n, a(n) for n = 1..18984
Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, 7373170279850, Math. Comp. 69 (2000), pp. 421-439. Appendix by I. Gusti Putu Purnaba.

Equivalent sequences for 1 way: A048926; 2 ways: A048927; 1 or more ways: A003328; 3 or more ways: A343704.

Cf. A003327.

Select[Range@2000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]==3&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,50)]#n
for pos in cwr(power_terms,5):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v == 3])#s
for x in range(len(rets)):
    print(rets[x])

A343702 Numbers that are the sum of five positive cubes in two or more ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710, 712, 719, 729, 731, 738, 745, 752, 759, 764, 766, 771, 773, 778, 785

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A048927:
766 = 1^3 + 1^3 + 2^3 + 3^3 + 9^3
    = 1^3 + 4^3 + 4^3 + 5^3 + 8^3
    = 2^3 + 2^3 + 4^3 + 7^3 + 7^3.
So 766 is a term, but not a term of A048927.

			227 = 1^3 + 1^3 + 1^3 + 2^3 + 6^3
    = 2^3 + 3^3 + 4^3 + 4^3 + 4^3
so 227 is a term of this sequence.

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A003328, A025406, A048927, A343704, A344238, A344795, A345511.

Select[Range@1000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]>1&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,50)]#n
for pos in cwr(power_terms,5):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 2])#s
for x in range(len(rets)):
    print(rets[x])

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

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)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

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).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

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

A046040 Numbers that are the sum of 6 but no fewer positive cubes.

Original entry on oeis.org

6, 13, 20, 34, 39, 41, 46, 48, 53, 58, 60, 69, 76, 79, 84, 86, 95, 98, 102, 104, 105, 110, 117, 121, 123, 124, 132, 139, 147, 151, 158, 165, 170, 173, 177, 184, 196, 202, 203, 210, 215, 221, 222, 228, 235, 236, 242, 247, 249, 263, 265, 268, 273, 275, 284, 287

Offset: 1

Eric W. Weisstein

According to the McCurley article, it is conjectured that there are exactly 3922 terms of which the largest is a(3922) = 1290740.

T. D. Noe, Table of n, a(n) for n=1..3922
Jan Bohman and Carl-Erik Froberg, Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
K. S. McCurley, An effective seven-cube theorem, J. Number Theory, 19 (1984), 176-183.
Eric Weisstein's World of Mathematics, Cubic Number.
Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for sequences related to sums of cubes

Cf. A000578, A003325, A003072, A003327, A003328, A018890, A018889.

Select[Range[300], (pr = PowersRepresentations[#, 6, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

Corrected by Arlin Anderson (starship1(AT)gmail.com).

A047704 Numbers that are the sum of 5 but no fewer positive cubes.

Original entry on oeis.org

5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 68, 71, 75, 78, 83, 90, 94, 96, 97, 101, 109, 115, 116, 120, 131, 138, 143, 146, 150, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 194, 195, 199, 201, 207, 208, 209, 213, 214, 220, 227, 234, 241, 246, 248

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

nonn

This sequence is conjectured to have a finite number of terms. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A003328, A002376.

Select[Range[300], (pr = PowersRepresentations[#, 5, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

A057906 Positive integers that are not the sum of exactly five positive cubes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 32, 34, 35, 36, 37, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 58, 60, 61, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 81, 84, 85, 86, 87, 88, 91, 92

Offset: 1

Eric W. Weisstein

It appears that this sequence has 5439 terms, the last of which is 1290740. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..5439
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. p. 5.
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A048926 (numbers that are the sum of five positive cubes in exactly 1 way)

Cf. A003328 (Complement)

max = 100; Complement[Range[max], Select[Range[max], Count[ PowersRepresentations[#, 5, 3], r_ /; FreeQ[r, 0]] == 1 &]] (* Jean-François Alcover, Oct 23 2012 *)

A025458 Number of partitions of n into 5 positive cubes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 0

David W. Wilson

nonn

a(n) > 2 at n= 766, 810, 827, 829, 865, 883, 981, 1018, 1025, 1044,... - R. J. Mathar, Sep 15 2015

The first term > 1 is a(157) = 2. - Michel Marcus, Apr 25 2019

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of cubes

Column 5 of A320841, which cross-references the equivalent sequences for other numbers of positive cubes.

Positions of values: A057906 (0), A003328 (nonzero), A048926 (1), A048927 (2), A343705 (3), A344035 (4).

A025458 := proc(n)
    local a,x,y,z,u,vcu ;
    a := 0 ;
    for x from 1 do
        if 5*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+4*y^3 > n then
                break;
            end if;
            for z from y do
                if x^3+y^3+3*z^3 > n then
                    break;
                end if;
                for u from z do
                    if x^3+y^3+z^3+2*u^3 > n then
                        break;
                    end if;
                    vcu := n-x^3-y^3-z^3-u^3 ;
                    if isA000578(vcu) then
                        a := a+1 ;
                    end if;
                end do:
            end do:
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015

a[n_] := IntegerPartitions[n, {5}, Range[n^(1/3) // Ceiling]^3] // Length;
a /@ Range[0, 157] (* Jean-François Alcover, Jun 20 2020 *)

a(n) = [x^n y^5] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Second offset from Michel Marcus, Apr 25 2019

cp's OEIS Frontend

A004814 Numbers that are the sum of 3 positive 11th powers.

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A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

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A048926 Numbers that are the sum of 5 positive cubes in exactly 1 way.

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A343705 Numbers that are the sum of five positive cubes in exactly three ways.

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A343702 Numbers that are the sum of five positive cubes in two or more ways.

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A003386 Numbers that are the sum of 8 nonzero 8th powers.

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A046040 Numbers that are the sum of 6 but no fewer positive cubes.

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A047704 Numbers that are the sum of 5 but no fewer positive cubes.

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