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

Author

David W. Wilson

Keywords

• nonn

Links

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

Crossrefs

Cf. A003329, A048929, A048930, A048931.

Programs

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 *)

Formula

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