A057907 -id:A057907 - cp's OEIS frontend

A003329 Numbers that are the sum of 6 positive cubes.

6, 13, 20, 27, 32, 34, 39, 41, 46, 48, 53, 58, 60, 65, 67, 69, 72, 76, 79, 83, 84, 86, 90, 91, 95, 97, 98, 102, 104, 105, 109, 110, 116, 117, 121, 123, 124, 128, 130, 132, 135, 136, 137, 139, 142, 143, 144, 146, 147, 151, 153, 154, 156, 158, 160, 161, 162, 163, 165, 170

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

			From _David A. Corneth_, Aug 01 2020: (Start)
1647 is in the sequence as 1647 = 3^3 + 3^3 + 5^3 + 5^3 +  7^3 + 10^3.
3319 is in the sequence as 3319 = 5^3 + 5^3 + 5^3 + 6^3 + 10^3 + 12^3.
4038 is in the sequence as 4038 = 3^3 + 3^3 + 6^3 + 8^3 +  8^3 + 14^3. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A048929, A048930, A048931.
Cf. A057907 (Complement)
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
  A000404 (2, 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).

max = 200; cmax = Ceiling[(max - 5)^(1/3)]; cc = Array[c, 6]; iter = Sequence @@ Transpose[ {cc, Join[{1}, Most[cc]], Table[cmax, {6}]}]; Union[ Reap[ Do[ a = Total[cc^3]; If[a <= max, Sow[a]], Evaluate[iter]]][[2, 1]]] (* Jean-François Alcover, Oct 23 2012 *)

(A003329_upto(N,k=6,m=3)=[i|i<-[1..#N=sum(n=1,sqrtnint(N,m), 'x^n^m, O('x^N))^k], polcoef(N,i)])(200) \\ M. F. Hasler, Aug 02 2020

from collections import Counter
from itertools import combinations_with_replacement as multi_combs
def aupto(lim):
  c = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
  s = filter(lambda x: x<=lim, (sum(mc) for mc in multi_combs(c, 6)))
  counts = Counter(s)
  return sorted(k for k in counts)
print(aupto(170)) # Michael S. Branicky, Jun 13 2021

More terms from Eric W. Weisstein

A048929 Numbers that are the sum of 6 positive cubes in exactly 1 way.

Original entry on oeis.org

6, 13, 20, 27, 32, 34, 39, 41, 46, 48, 53, 58, 60, 65, 67, 69, 72, 76, 79, 83, 84, 86, 90, 91, 95, 97, 98, 102, 104, 105, 109, 110, 116, 117, 121, 123, 124, 128, 130, 132, 135, 136, 137, 139, 142, 143, 144, 146, 147, 151, 153, 154, 156, 160, 161, 162, 163, 170

Offset: 1

Eric W. Weisstein

It appears that this sequence has 841 terms, the last of which is 19417. This means that all numbers greater than 19417 can be written as the sum of six positive cubes in at least two ways. - T. D. Noe, Dec 13 2006

T. D. Noe, Table of n, a(n) for n=1..841
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003329, A048930, A048931.

Cf. A057907 (numbers not the sum of six positive cubes)

Select[ Range[200], Length[ Select[ PowersRepresentations[#, 6, 3], And @@ (Positive /@ #) &]] == 1 &] (* Jean-François Alcover, Oct 25 2012 *)

from collections import Counter
from itertools import combinations_with_replacement as multi_combs
def aupto(lim):
  c = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
  s = filter(lambda x: x<=lim, (sum(mc) for mc in multi_combs(c, 6)))
  counts = Counter(s)
  return sorted(k for k in counts if counts[k]==1)
print(aupto(20000)) # Michael S. Branicky, Jun 13 2021

A069137 Numbers which are sums of neither 1, 2, 3, 4, 5 or 6 nonnegative cubes.

Original entry on oeis.org

7, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356, 363, 364

Offset: 1

N. J. A. Sloane, Apr 08 2002; edited Sep 15 2006

nonn

Sequence is conjectured to be finite.

			Numbers which need at least seven terms to represent them as a sum of positive cubes: 14=8+1+1+1+1+1+1.

Bohman, Jan and Froberg, Carl-Erik; Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.

T. D. Noe, Table of n, a(n) for n=1..138
Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau; appendix by I. Gusti Putu Purnaba, 7373170279850, Math. Comp. 69 (2000), 421-439.
Index entries for sequences related to sums of cubes

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A000578, A085334.

Natural numbers remaining if union of A003325, A003072, A003327, A003328, A003329 and A000578 sets were deleted. Remark: this sequence itself does not include cubes, in contrast to A085334.

A085334 Numbers which are neither sums of 2,3,4,5 or that of 6 nonnegative cubes.

Original entry on oeis.org

1, 7, 8, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 125, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356

Offset: 1

Labos Elemer, Jul 07 2003

nonn

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A069137.

Remaining set of all natural numbers if union of A003325, A003072, A003327, A003328 and A003329 sets were deleted.Remark: this sequence includes those cubes too, which are sums of 7 or more cubes.

A158794 Multiples of 4 which are not the sum of seven nonnegative cubes.

Original entry on oeis.org

212, 364, 420, 428

Offset: 1

Jonathan Vos Post, Mar 26 2009

Boklan and Elkies: It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.

Elkies [2010]: It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is congruent to 2 mod 4. This result, together with a recent proof for 4|N, shows that the conjecture is true for all even N. - Jonathan Vos Post, Sep 22 2010

U. V. Linnik: On the representation of large numbers as sums of seven cubes. Rec. Math. [=Mat. Sbornik] N.S. 12(54) (1943), 218-224.
L. E. Dickson: All integers except 23 and 239 are the sums of 8 cubes. Bull. Amer. Math. Soc. 45 (1939), 588-591.

E. Waring: Meditationes Algebraicae [3rd ed. (1782)]: an English translation of the work of Edward Waring, edited and translated from the Latin by Dennis Weeks. Providence, RI: Amer. Math. Soc., 1991. [MR1146921]
Kent D. Boklan and Noam D. Elkies, Every multiple of 4 except 212, 364, 420, and 428 is the sum of seven cubes, arXiv:0903.4503, Mar 26, 2009.
Noam D. Elkies, Every even number greater than 454 is the sum of seven cubes, Sep 21, 2010. [From _Jonathan Vos Post_, Sep 22 2010]

Cf. A000578, A003072, A003325, A003327, A003328, A003329, A057907, A122730.

Subsequence of A018888. - Charles R Greathouse IV, Apr 16 2010

Definition corrected by Jonathan Sondow, Mar 14 2014

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A003329 Numbers that are the sum of 6 positive cubes.

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A048929 Numbers that are the sum of 6 positive cubes in exactly 1 way.

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A069137 Numbers which are sums of neither 1, 2, 3, 4, 5 or 6 nonnegative cubes.

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A085334 Numbers which are neither sums of 2,3,4,5 or that of 6 nonnegative cubes.

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A158794 Multiples of 4 which are not the sum of seven nonnegative cubes.

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