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.
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Dec 10 2020
STATUS
approved