A339621 Sum of Fibonacci divisors of n^2 + 1.

1, 3, 6, 8, 1, 16, 1, 8, 19, 3, 1, 3, 6, 42, 1, 3, 1, 8, 19, 3, 1, 50, 6, 8, 1, 3, 1, 8, 6, 3, 1, 16, 6, 8, 103, 3, 1, 8, 6, 3, 1, 3, 6, 8, 14, 3, 1, 55, 6, 3, 1, 3, 6, 8, 1, 126, 1, 21, 6, 3, 14, 3, 6, 8, 1, 3, 1, 8, 6, 3, 391, 3, 6, 21, 1, 3, 1, 8, 6, 3, 1, 37

Offset: 0

Author

Michel Lagneau, Dec 10 2020

Keywords

• nonn

Comments

A Fibonacci divisor of a number k is a Fibonacci number that divides k. (The divisor 1 is only counted once.)

For n < 2*10^5, the subsequence of primes begins by 3, 19, 37, 97, 103, 131, 139, 239, 241, 283, 359, 487, 631, ...

The Fibonacci numbers of the sequence are 1, 3, 8, 21, 55, 144, 377, ...

Conjecture: If the sum of the Fibonacci divisors of m^2 + 1 is a Fibonacci number, then this number belongs to the sequence A001906(n) = F(2n) where F(n) is the Fibonacci sequence.

The sequence giving the least k such that the sum of Fibonacci divisors of k^2 + 1 is equal to F(2*n) for n > 0 begins with: 0, 1, 3, 57, 47, 15007, 1679553, ...

Examples

			a(3) = 8 because the divisors of 3^2 + 1 = 10 are {1, 2, 5, 10}, and the sum of the Fibonacci divisors is 1 + 2 + 5 = 8.
		

Links

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

Crossrefs

Cf. A000045, A001906, A002522, A005092, A005574, A339461.

Programs

Maple

a:= n-> add(`if`(issqr(5*d^2+4) or issqr(5*d^2-4), d, 0)
, d=numtheory[divisors](n^2+1)):seq(a(n), n=0..100);

Mathematica

Array[DivisorSum[#^2 + 1, # &, AnyTrue[Sqrt[5 #^2 + 4 {-1, 1}], IntegerQ] &] &, 82, 0] (* Michael De Vlieger, Dec 10 2020 *)

PARI

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
a(n) = sumdiv(n^2+1, d, if (isfib(d), d)); \\ Michel Marcus, Dec 10 2020

Formula

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

a(n) = 3 when n^2 + 1 = 2*p, p prime and non-Fibonacci number.

a(n) = A005092(A002522(n)). - Michel Marcus, Aug 10 2022