A025456 Number of partitions of n into 3 positive cubes.

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

Offset

0,252

Comments

If A025455(n) > 0 then a(n + k^3) > 0 for k>0; a(A119977(n))>0; a(A003072(n))>0. - Reinhard Zumkeller, Jun 03 2006

a(A057904(n))=0; a(A003072(n))>0; a(A025395(n))=1; a(A008917(n))>1; a(A025396(n))=2. - Reinhard Zumkeller, Apr 23 2009

The first term > 1 is a(251) = 2. - Michel Marcus, Apr 23 2019

Hardy and Littlewood conjectured (K hypothesis) that the number of representations of n as the sum of k many k-th powers is less than n^o(1). However for k=3 (this sequence) the hypothesis has been disproven by Mahler in 1936. - Elijah Beregovsky, Oct 02 2025

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Thomas Bloom, Problem #322, Erdős Problems.

S. Chowla and S. S. Pillai, The number of representations of a number as a sum of non-negative nth powers, Quart. J. Math. Oxford, 7 (1936) 56-59.

Kurt Mahler, Note on Hypothesis K of Hardy and Littlewood, J. London Math. Soc. (1936), 136-138.

Index entries for sequences related to sums of cubes

Formula

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

There exists a constant c such that a(n) >> n^(c/log log n) (proved independently by Erdős and Chowla&Pillai). - Elijah Beregovsky, Oct 02 2025

Maple

A025456 := proc(n)
    local a, x, y, zcu ;
    a := 0 ;
    for x from 1 do
        if 3*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+2*y^3 > n then
                break;
            end if;
            zcu := n-x^3-y^3 ;
            if isA000578(zcu) then
                a := a+1 ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015

Mathematica

a[n_] := Count[ PowersRepresentations[n, 3, 3], pr_List /; FreeQ[pr, 0]]; Table[a[n], {n, 0, 107}] (* Jean-François Alcover, Oct 31 2012 *)

Prog

(PARI) a(n)=sum(a=sqrtnint(n\3, 3), sqrtnint(n, 3), sum(b=1, a, my(C=n-a^3-b^3, c); ispower(C, 3, &c)&&0<c&&c<=b)) \\ Charles R Greathouse IV, Jun 26 2013

Crossrefs

Least inverses are A025418.

Cf. A025455, A003108, A003072 (1 or more ways), A008917 (two or more ways), A025395-A025398.

Sequence in context: A328981 A369070 A024360 * A288314 A285963 A024889

Adjacent sequences: A025453 A025454 A025455 * A025457 A025458 A025459

Keyword

nonn

Author

David W. Wilson

Extensions

Second offset from Michel Marcus, Apr 23 2019

Status

approved