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

1,3

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A353843 A090002 A061298 * A276468 A002126 A350815

Adjacent sequences: A385901 A385902 A385903 * A385905 A385906 A385907

Keyword

nonn

Author

Felix Huber, Jul 21 2025

Status

approved