A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 2, 2, 1, 4, 1, 3, 2, 2, 2, 3, 1, 2, 3, 4, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 4, 1, 4, 4

Offset: 1

Author

Antti Karttunen, Oct 09 2017

Keywords

• nonn

Examples

			For n = 55, its proper divisors are [1, 5, 11], of which only two, namely 1 and 5 are in A000045, thus a(55) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10946

Crossrefs

Cf. A000045, A005086, A010056, A079586, A293436.

Cf. also A293433.

Programs

Mathematica

With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

PARI

A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ This function from Charles R Greathouse IV, Jul 30 2012
A293435(n) = sumdiv(n,d,(dA010056(d));

Formula

a(n) = Sum_{d|n, dA010056(d).

a(n) = A005086(n) - A010056(n).

G.f.: Sum_{k>=2} x^(2*Fibonacci(k)) / (1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Apr 14 2021

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A079586 - 1 = 2.359885... . - Amiram Eldar, Jul 05 2025