A339461 Number of Fibonacci divisors of n^2 + 1.

1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 2, 2, 4, 1, 2, 1, 3, 3, 2, 1, 4, 2, 3, 1, 2, 1, 3, 2, 2, 1, 3, 2, 3, 3, 2, 1, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 5, 2, 2, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 3, 2, 2, 4, 1, 2, 1, 3, 2, 2, 1, 3, 2, 4, 1, 2, 2

Offset: 0

Author

Michel Lagneau, Dec 06 2020

Keywords

• nonn

Examples

			a(13) = 4 because the divisors of 13^2 + 1 = 170 are {1, 2, 5, 10, 17, 34, 85, 170} with 4 Fibonacci divisors: 1, 2, 5 and 34.
		

Links

• Michel Marcus, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000045, A002522, A005086, A005478, A005574, A128428, A338762, A338794.

Programs

Maple

with(numtheory):with(combinat,fibonacci):nn:=100:F:={}:
for k from 1 to nn do:
  F:=F union {fibonacci(k)}:
od:
   for n from 0 to 90 do:
    f:=n^2+1:d:=divisors(f):
    lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):
   od:

Mathematica

Array[DivisorSum[#^2 + 1, 1 &, Or @@ Map[IntegerQ@ Sqrt[#] &, 5 #^2 + 4 {-1, 1}] &] &, 105, 0] (* Michael De Vlieger, Dec 07 2020 *)

PARI

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
a(n) = sumdiv(n^2+1, d, isfib(d)); \\ Michel Marcus, Dec 06 2020

Formula

a(A005574(n)) = 1 for n > 2.

a(n) = A005086(A002522(n)). - Michel Marcus, Dec 06 2020