A018890 -id:A018890 - cp's OEIS frontend

A002376 Least number of positive cubes needed to sum to n.

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 1, 2, 3, 4, 5, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 5, 6, 7, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 3, 4, 5, 6, 7, 6, 7, 4, 4, 5, 2, 3, 4, 5, 6, 5, 5, 6, 3, 4, 5, 6, 7, 6, 6

Offset: 1

N. J. A. Sloane

No terms are greater than 9, see A002804. - Charles R Greathouse IV, Aug 01 2013

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. R. Zornow, De compositione numerorum e cubis integris positivus, J. Reine Angew. Math., 14 (1835), 276-280.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A003325 (numbers requiring 2 cubes), A047702 (numbers requiring 3 cubes), A047703 (numbers requiring 4 cubes), A047704 (numbers requiring 5 cubes), A046040 (numbers requiring 6 cubes), A018890 (numbers requiring 7 cubes), A018888 (numbers requiring 8 or 9 cubes), A055401 (cubes needed by greedy algorithm).

f:= proc(n) option remember;
  min(seq(procname(n - i^3)+1, i=1..floor(n^(1/3))))
end proc:
f(0):= 0:
map(f, [$1..100]); # Robert Israel, Jun 30 2017

CubesCnt[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 3]] == 0, k++]; k]; Array[CubesCnt, 100] (* T. D. Noe, Apr 01 2011 *)

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A002376(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,3,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

The g.f. conjectured by Simon Plouffe in his 1992 dissertation,

-(-1-z-z^2-z^3-z^4-z^5-z^6+6*z^7)/(z+1)/(z^2+1)/(z^4+1)/(z-1)^2, is incorrect: the first wrong coefficient is that of z^26. - Robert Israel, Jun 30 2017

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

A181402 Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 10, 73, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 1

Martin Renner, Jan 28 2011

An unpublished result of Deshouillers-Hennecart-Landreau, combined with Lemma 3 from Bertault, Ramaré, & Zimmermann implies that a(4)-a(34) are all 121. Probably a(n) = 121 for all n > 3. - Charles R Greathouse IV, Jan 23 2014

F. Bertault, O. Ramaré, and P. Zimmermann, On sums of seven cubes, Math. Comp. 68 (1999), pp. 1303-1310.
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A018890.

Cf. A002283, A061439, A130130, A181375, A181377, A181379, A181381, A181400, A181404.

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + a(n) + A181404(n) + A130130(n) = A002283(n).
Conjectured g.f.: x*(1+9*x+63*x^2+48*x^3)/(1-x). - Colin Barker, May 04 2012
Conjectured e.g.f.: 121*(exp(x) - 1) - 120*x - 111*x^2/2 - 8*x^3. - Stefano Spezia, May 21 2024

a(5)-a(7) from Lars Blomberg, May 04 2011

a(8)-a(34) from Charles R Greathouse IV, Jan 23 2014

A181403 Total number of n-digit numbers requiring 7 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 9, 63, 48

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + a(n) + A181405(n) + A171386(n) = A052268(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A018890.

a(n) = A181402(n) - A181402(n-1).

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).

A245229 Primes that are the sum of 7 cubes and no fewer.

Original entry on oeis.org

7, 47, 61, 103, 113, 211, 223, 229, 311, 337, 401, 419, 491, 787, 1021, 1453, 1489, 1697, 2039, 3659, 4703, 5279

Offset: 1

Rafael F. Farias, Jul 13 2014

Intersection of A018890 and A000040.

If, as is conjectured, the last term of A018890 is 8042, there are no more terms than those shown. - Robert Israel, Jul 14 2014

			a(1) = 7 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3.
a(2) = 47 = 3^3 + 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3.
a(3) = 61 = 3^3 + 2^3 + 2^3 + 2^3 + 2^3 + 1^3 + 1^3.
a(4) = 103 = 4^3 + 3^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3.

Cf. A000578, A018890, A000040.

for n from 1 to 10^4 do
  m:= floor(n^(1/3));
  if m^3 = n then M[n]:= 1
  else
    M[n]:= 1 + min(seq(M[n-j^3],j=1..m));
  fi
od:
select(n -> M[n]=7 and isprime(n), [$1..10^4]); # Robert Israel, Jul 14 2014

cp's OEIS Frontend

A002376 Least number of positive cubes needed to sum to n.

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A181402 Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

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A181403 Total number of n-digit numbers requiring 7 positive cubes in their representation as sum of cubes.

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

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A245229 Primes that are the sum of 7 cubes and no fewer.

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Maple