cp's OEIS Frontend

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

%I A385904 #14 Jul 28 2025 09:01:13
%S A385904 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,
%T A385904 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,
%U A385904 11,1,21,1,4,2,17,1,235,1,1,9,7,1,20,1,64,6,1
%N A385904 a(n) is the number of nonempty subsets of the divisors of n that sum to a perfect square.
%H A385904 Felix Huber, <a href="/A385904/b385904.txt">Table of n, a(n) for n = 1..10000</a>
%F A385904 a(p) = 1 for primes p != 3.
%e A385904 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.
%p A385904 with(NumberTheory):
%p A385904 A385904:=proc(n)
%p A385904     local b,l,j;
%p A385904     l:=[(Divisors(n))[]]:
%p A385904     b:=proc(m,i)
%p A385904         option remember;
%p A385904         `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
%p A385904 	end;
%p A385904     add(b(j^2,nops(l)),j=1..floor(sqrt(sigma(n))));
%p A385904 end:
%p A385904 seq(A385904(n),n=1..82);
%t A385904 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 *)
%o A385904 (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
%Y A385904 Cf. A000290, A005835, A027750, A119347, A119348, A237290, A385645, A385646.
%K A385904 nonn
%O A385904 1,3
%A A385904 _Felix Huber_, Jul 21 2025