A025457 Number of partitions of n into 4 positive cubes.

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

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

The first term > 1 is a(219) = 2. - Michel Marcus, Apr 23 2019

Links

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

Crossrefs

Cf. A003108, A025455, A025456, A025403-A025407, A003327, A025420 (greedy inverse).

Programs

Maple

N:= 100;
A:= Array(0..N);
for a from 1 to floor(N^(1/3)) do
  for b from a to floor((N-a^3)^(1/3)) do
     for c from b to floor((N-a^3-b^3)^(1/3)) do
        for d from c to floor((N-a^3-b^3-c^3)^(1/3)) do
          n:= a^3 + b^3 + c^3 + d^3;
          A[n]:= A[n]+1;
od od od od:
seq(A[n],n=0..N); # Robert Israel, Aug 18 2014
A025457 := proc(n)
    local a,x,y,z,ucu ;
    a := 0 ;
    for x from 1 do
        if 4*x^3 > n then
            return a;
        end if;
        for y from x do
            if x^3+3*y^3 > n then
                break;
            end if;
            for z from y do
                if x^3+y^3+2*z^3 > n then
                    break;
                end if;
                ucu := n-x^3-y^3-z^3 ;
                if isA000578(ucu) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc: # R. J. Mathar, Sep 15 2015

Mathematica

r[n_] := Reduce[0 < a <= b <= c <= d && n == a^3+b^3+c^3+d^3, {a, b, c, d}, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
Table[a[n], {n, 0, 107}] (* Jean-François Alcover, Feb 26 2019 *)

Formula

a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Extensions

Second offset from Michel Marcus, Apr 23 2019