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

%I A375731 #18 Sep 16 2024 23:58:36
%S A375731 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,
%T A375731 43,54,54,64,87,85,119,128,146,174,205,213,281,324,368,420,503,531,
%U A375731 688,760,837,992,1174,1252,1535,1705,1931,2236,2619,2821,3402,3769,4272
%N A375731 a(n) is the number of partitions of n having a square number of parts whose sum of squares is a square.
%F A375731 1 <= a(n) <= A240127(n).
%e A375731 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.
%p A375731 # first Maple program to calculate the sequence:
%p A375731 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);
%p A375731 # second Maple program to calculate the partitions:
%p A375731 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);
%o A375731 (PARI) a(n) = my(nb=0); forpart(p=n, if (issquare(#p) && issquare(norml2(Vec(p))), nb++)); nb; \\ _Michel Marcus_, Aug 30 2024
%Y A375731 Cf. A001156, A025428, A006456, A037444, A240127, A240128, A375732.
%K A375731 nonn
%O A375731 0,5
%A A375731 _Felix Huber_, Aug 28 2024