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

0,2

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, ...

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A181917 A157032 A295655 * A254604 A278498 A011334

Adjacent sequences: A339618 A339619 A339620 * A339622 A339623 A339624

Keyword

nonn

Author

Michel Lagneau, Dec 10 2020

Status

approved