A025459 - OEIS

%I #24 Jan 09 2023 07:41:18

%S 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 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 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 Number of partitions of n into 6 positive cubes.

%H Robert Israel, <a href="/A025459/b025459.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>

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

%p A025459 := proc(n)

%p     local a,x,y,z,u,v,wcu ;

%p     a := 0 ;

%p     for x from 1 do

%p         if 6*x^3 > n then

%p             return a;

%p         end if;

%p         for y from x do

%p             if x^3+5*y^3 > n then

%p                 break;

%p             end if;

%p             for z from y do

%p                 if x^3+y^3+4*z^3 > n then

%p                     break;

%p                 end if;

%p                 for u from z do

%p                     if x^3+y^3+z^3+3*u^3 > n then

%p                         break;

%p                     end if;

%p                     for v from u do

%p                         if x^3+y^3+z^3+u^3+2*v^3 > n then

%p                             break;

%p                         end if;

%p                         wcu := n-x^3-y^3-z^3-u^3-v^3 ;

%p                         if isA000578(wcu) then

%p                             a := a+1 ;

%p                         end if;

%p                     end do:

%p                 end do:

%p             end do:

%p         end do:

%p     end do:

%p end proc: # _R. J. Mathar_, Sep 15 2015

%p # Alternative:

%p N:= 200:

%p G:= mul(1/(1-y*x^(k^3)),k=1..floor(N^(1/3))):

%p C6:= coeff(series(G,y,7),y,6):

%p S:= series(C6,x,N+1):

%p seq(coeff(S,x,i),i=0..N); # _Robert Israel_, May 10 2020

%t a[n_] := Count[PowersRepresentations[n, 6, 3], pr_List /; FreeQ[pr, 0]];

%t a /@ Range[0, 200] (* _Jean-François Alcover_, Jun 22 2020 *)

%t 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 Cf. A003329, A048929, A048930, A048931.

%K nonn

%O 0,159

%A _David W. Wilson_