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A025458 Number of partitions of n into 5 positive cubes.

%I A025458 #25 May 16 2021 01:55:58
%S A025458 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,
%T A025458 0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,
%U A025458 1,0,0,1,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0
%N A025458 Number of partitions of n into 5 positive cubes.
%C A025458 a(n) > 2 at n= 766, 810, 827, 829, 865, 883, 981, 1018, 1025, 1044,... - _R. J. Mathar_, Sep 15 2015
%C A025458 The first term > 1 is a(157) = 2. - _Michel Marcus_, Apr 25 2019
%H A025458 Robert Israel, <a href="/A025458/b025458.txt">Table of n, a(n) for n = 0..10000</a>
%H A025458 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A025458 a(n) = [x^n y^5] Product_{k>=1} 1/(1 - y*x^(k^3)). - _Ilya Gutkovskiy_, Apr 23 2019
%p A025458 A025458 := proc(n)
%p A025458     local a,x,y,z,u,vcu ;
%p A025458     a := 0 ;
%p A025458     for x from 1 do
%p A025458         if 5*x^3 > n then
%p A025458             return a;
%p A025458         end if;
%p A025458         for y from x do
%p A025458             if x^3+4*y^3 > n then
%p A025458                 break;
%p A025458             end if;
%p A025458             for z from y do
%p A025458                 if x^3+y^3+3*z^3 > n then
%p A025458                     break;
%p A025458                 end if;
%p A025458                 for u from z do
%p A025458                     if x^3+y^3+z^3+2*u^3 > n then
%p A025458                         break;
%p A025458                     end if;
%p A025458                     vcu := n-x^3-y^3-z^3-u^3 ;
%p A025458                     if isA000578(vcu) then
%p A025458                         a := a+1 ;
%p A025458                     end if;
%p A025458                 end do:
%p A025458             end do:
%p A025458         end do:
%p A025458     end do:
%p A025458 end proc: # _R. J. Mathar_, Sep 15 2015
%t A025458 a[n_] := IntegerPartitions[n, {5}, Range[n^(1/3) // Ceiling]^3] // Length;
%t A025458 a /@ Range[0, 157] (* _Jean-François Alcover_, Jun 20 2020 *)
%Y A025458 Column 5 of A320841, which cross-references the equivalent sequences for other numbers of positive cubes.
%Y A025458 Positions of values: A057906 (0), A003328 (nonzero), A048926 (1), A048927 (2), A343705 (3), A344035 (4).
%K A025458 nonn
%O A025458 0,158
%A A025458 _David W. Wilson_
%E A025458 Second offset from _Michel Marcus_, Apr 25 2019