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

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, 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, 47, 60, 54, 77, 87, 83, 89, 88, 104, 131, 156, 170, 202, 208, 220, 241

Offset: 0

Author

Felix Huber, Aug 28 2024

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A003108, A025457, A240127, A240128, A259792, A375731.

Programs

Maple

# first Maple program to calculate the sequence:
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);
# second Maple program to calculate the partitions:
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);

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

Formula

1 <= a(n) <= A240128(n).