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A375732 a(n) is the number of partitions of n having a cube number of parts whose sum of cubes is a cube.

%I A375732 #15 Sep 16 2024 23:58:32
%S A375732 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,2,2,1,1,1,1,1,2,2,2,5,2,2,1,2,2,2,
%T A375732 4,5,9,4,5,2,6,9,9,13,12,16,8,10,8,13,19,20,26,23,23,22,22,30,38,45,
%U A375732 47,60,54,77,87,83,89,88,104,131,156,170,202,208,220,241
%N A375732 a(n) is the number of partitions of n having a cube number of parts whose sum of cubes is a cube.
%F A375732 1 <= a(n) <= A240128(n).
%e A375732 a(37) counts the 4 partitions [1, 1, 1, 2, 6, 8, 9, 9] with 8 = 2^3 parts and 1^3 + 1^3 + 1^3 + 2^3 + 6^3 + 8^3 + 9^3 + 9^3 = 13^3, [1, 1, 2, 4, 4, 6, 8, 11] with 8 = 2^3 parts and 1^3 + 1^3 + 2^3 + 4^3 + 4^3 + 6^3 + 8^3 + 11^3 = 13^3, [1, 1, 1, 2, 2, 2, 10, 18] with 8 = 2^3 parts and 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 10^3 + 18^3 = 19^3, [37] with 1 = 1^3 part and 37^3 = 37^3.
%p A375732 # first Maple program to calculate the sequence:
%p A375732 A375732:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if type(root(numelems(i),3),integer) and type(root(add(i[j]^3,j=1..nops(i)),3), integer) then a:=a+1 fi od; return a end proc; seq(A375732(n),n=0..75);
%p A375732 # second Maple program to calculate the partitions:
%p A375732 A375732part:=proc(n) local L,i,j; L:=[]; for i in combinat:-partition(n) do if type(root(numelems(i),3),integer) and type(root(add(i[j]^3,j=1..nops(i)), 3),integer) then L:=[op(L),i] fi od; return op(L); end proc; A375732part(37);
%o A375732 (PARI) a(n) = my(nb=0); forpart(p=n, if (ispower(#p,3) && ispower(sum(k=1, #p, p[k]^3),3), nb++)); nb; \\ _Michel Marcus_, Sep 01 2024
%Y A375732 Cf. A003108, A025457, A240127, A240128, A259792, A375731.
%K A375732 nonn
%O A375732 0,9
%A A375732 _Felix Huber_, Aug 28 2024