A385904 a(n) is the number of nonempty subsets of the divisors of n that sum to a perfect square.

1, 1, 2, 2, 1, 4, 1, 3, 3, 2, 1, 11, 1, 3, 4, 5, 1, 9, 1, 9, 3, 3, 1, 27, 2, 2, 4, 8, 1, 27, 1, 7, 3, 2, 2, 49, 1, 1, 3, 22, 1, 21, 1, 7, 8, 3, 1, 77, 2, 5, 2, 4, 1, 22, 2, 21, 2, 1, 1, 248, 1, 2, 7, 11, 1, 21, 1, 4, 2, 17, 1, 235, 1, 1, 9, 7, 1, 20, 1, 64, 6, 1

Offset: 1

Author

Felix Huber, Jul 21 2025

Keywords

• nonn

Examples

			a(6) = 4 because exactly the 4 nonempty subsets {1}, {1, 3}, {1, 2, 6} and {3, 6} of the divisors of 6 sum to a perfect square: 1 = 1^2, 1 + 3 = 2^2, 1 + 2 + 6 = 3^2.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A005835, A027750, A119347, A119348, A237290, A385645, A385646.

Programs

Maple

with(NumberTheory):
A385904:=proc(n)
    local b,l,j;
    l:=[(Divisors(n))[]]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
	end;
    add(b(j^2,nops(l)),j=1..floor(sqrt(sigma(n))));
end:
seq(A385904(n),n=1..82);

Mathematica

a[n_]:=Module[{nb = 0, d = Divisors[n]},Length[Select[Subsets[d],IntegerQ[Sqrt[Total[#]]]&]]]-1;Array[a,82] (* James C. McMahon, Jul 27 2025 *)

PARI

a(n) = my(nb=0, d=divisors(n)); forsubset(#d, s, nb+=issquare(sum(i=1, #s, d[s[i]]))); nb-1; \\ Michel Marcus, Jul 22 2025

Formula

a(p) = 1 for primes p != 3.