A025459 Number of partitions of n into 6 positive cubes.

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

Offset

0,159

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Index entries for sequences related to sums of cubes

Formula

a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Maple

A025459 := proc(n)
    local a, x, y, z, u, v, wcu ;
    a := 0 ;
    for x from 1 do
        if 6*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+5*y^3 > n then
                break;
            end if;
            for z from y do
                if x^3+y^3+4*z^3 > n then
                    break;
                end if;
                for u from z do
                    if x^3+y^3+z^3+3*u^3 > n then
                        break;
                    end if;
                    for v from u do
                        if x^3+y^3+z^3+u^3+2*v^3 > n then
                            break;
                        end if;
                        wcu := n-x^3-y^3-z^3-u^3-v^3 ;
                        if isA000578(wcu) then
                            a := a+1 ;
                        end if;
                    end do:
                end do:
            end do:
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015
# Alternative:
N:= 200:
G:= mul(1/(1-y*x^(k^3)), k=1..floor(N^(1/3))):
C6:= coeff(series(G, y, 7), y, 6):
S:= series(C6, x, N+1):
seq(coeff(S, x, i), i=0..N); # Robert Israel, May 10 2020

Mathematica

a[n_] := Count[PowersRepresentations[n, 6, 3], pr_List /; FreeQ[pr, 0]];
a /@ Range[0, 200] (* Jean-François Alcover, Jun 22 2020 *)
Table[Count[IntegerPartitions[n, {6}], _?(AllTrue[Surd[#, 3], IntegerQ]&)], {n, 0, 110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 06 2021 *)

Crossrefs

Cf. A003329, A048929, A048930, A048931.

Sequence in context: A373256 A286996 A044937 * A079365 A037822 A144600

Adjacent sequences: A025456 A025457 A025458 * A025460 A025461 A025462

Keyword

nonn

Author

David W. Wilson

Status

approved