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A025459 Number of partitions of n into 6 positive cubes.

%I A025459 #24 Jan 09 2023 07:41:18
%S A025459 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,
%T A025459 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,
%U A025459 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
%N A025459 Number of partitions of n into 6 positive cubes.
%H A025459 Robert Israel, <a href="/A025459/b025459.txt">Table of n, a(n) for n = 0..10000</a>
%H A025459 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A025459 a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^(k^3)). - _Ilya Gutkovskiy_, Apr 23 2019
%p A025459 A025459 := proc(n)
%p A025459     local a,x,y,z,u,v,wcu ;
%p A025459     a := 0 ;
%p A025459     for x from 1 do
%p A025459         if 6*x^3 > n then
%p A025459             return a;
%p A025459         end if;
%p A025459         for y from x do
%p A025459             if x^3+5*y^3 > n then
%p A025459                 break;
%p A025459             end if;
%p A025459             for z from y do
%p A025459                 if x^3+y^3+4*z^3 > n then
%p A025459                     break;
%p A025459                 end if;
%p A025459                 for u from z do
%p A025459                     if x^3+y^3+z^3+3*u^3 > n then
%p A025459                         break;
%p A025459                     end if;
%p A025459                     for v from u do
%p A025459                         if x^3+y^3+z^3+u^3+2*v^3 > n then
%p A025459                             break;
%p A025459                         end if;
%p A025459                         wcu := n-x^3-y^3-z^3-u^3-v^3 ;
%p A025459                         if isA000578(wcu) then
%p A025459                             a := a+1 ;
%p A025459                         end if;
%p A025459                     end do:
%p A025459                 end do:
%p A025459             end do:
%p A025459         end do:
%p A025459     end do:
%p A025459 end proc: # _R. J. Mathar_, Sep 15 2015
%p A025459 # Alternative:
%p A025459 N:= 200:
%p A025459 G:= mul(1/(1-y*x^(k^3)),k=1..floor(N^(1/3))):
%p A025459 C6:= coeff(series(G,y,7),y,6):
%p A025459 S:= series(C6,x,N+1):
%p A025459 seq(coeff(S,x,i),i=0..N); # _Robert Israel_, May 10 2020
%t A025459 a[n_] := Count[PowersRepresentations[n, 6, 3], pr_List /; FreeQ[pr, 0]];
%t A025459 a /@ Range[0, 200] (* _Jean-François Alcover_, Jun 22 2020 *)
%t A025459 Table[Count[IntegerPartitions[n,{6}],_?(AllTrue[Surd[#,3],IntegerQ]&)],{n,0,110}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 06 2021 *)
%Y A025459 Cf. A003329, A048929, A048930, A048931.
%K A025459 nonn
%O A025459 0,159
%A A025459 _David W. Wilson_