A375731 a(n) is the number of partitions of n having a square number of parts whose sum of squares is a square.

1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 4, 2, 2, 7, 4, 4, 7, 7, 6, 9, 12, 9, 21, 21, 19, 26, 30, 32, 43, 54, 54, 64, 87, 85, 119, 128, 146, 174, 205, 213, 281, 324, 368, 420, 503, 531, 688, 760, 837, 992, 1174, 1252, 1535, 1705, 1931, 2236, 2619, 2821, 3402, 3769, 4272

Offset: 0

Author

Felix Huber, Aug 28 2024

Keywords

• nonn

Examples

			a(13) counts the 4 partitions [1, 1, 1, 1, 1, 1, 1, 3, 3] with 9 = 3^2 parts and 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 = 5^2, [1, 4, 4, 4] with 2^2 parts and 1^2 + 4^2 + 4^2 + 4^2 = 7^2, [2, 2, 4, 5] with 4 = 2^2 parts and 2^2 + 2^2 + 4^2 + 5^2 = 7^2, [13] with 1 = 1^2 part and 13^2 = 13^2.
		

Crossrefs

Cf. A001156, A025428, A006456, A037444, A240127, A240128, A375732.

Programs

Maple

# first Maple program to calculate the sequence:
A375731:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then a:=a+1 fi od; return a end proc; seq(A375731(n),n=0..63);
# second Maple program to calculate the partitions:
A375731part:=proc(n) local L,i,j;L:=[]; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then L:=[op(L),i] fi od; return op(L) end proc; A375731part(13);

PARI

a(n) = my(nb=0); forpart(p=n, if (issquare(#p) && issquare(norml2(Vec(p))), nb++)); nb; \\ Michel Marcus, Aug 30 2024

Formula

1 <= a(n) <= A240127(n).