A338762 Greatest prime Fibonacci divisor of F(n)^2 + 1 where F(n) is the n-th Fibonacci number, or 0 if no such prime factor exists.

2, 2, 5, 5, 13, 13, 5, 13, 89, 89, 233, 233, 89, 233, 1597, 1597, 5, 1597, 1597, 13, 28657, 28657, 13, 28657, 28657, 5, 514229, 514229, 2, 514229, 514229, 89, 13, 89, 89, 13, 233, 233, 0, 233, 433494437, 433494437, 5, 433494437, 2971215073, 2971215073, 13, 2971215073

Offset: 1

Author

Michel Lagneau, Nov 07 2020

Keywords

• nonn

Comments

a(n) = 0 for n = 39, 60, 69, 72, ... .

a(5385) has 1126 decimal digits. - Chai Wah Wu, Nov 19 2020

Examples

			a(6) = 13 because F(6)^2 + 1 = 8^2 + 1 = 65 = 5*13 and 13 is the greatest prime Fibonacci divisor.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..5384 (n = 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A000045, A005478, A245306.

Programs

Maple

a:= proc(n) local F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      while F[2]<=t do if isprime(F[2]) and irem(t, F[2])=0
        then m:=F[2] fi; F:= [F[2], F[1]+F[2]]
      od; m
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 07 2020

Mathematica

a[n_] := Module[{F, m, t}, F = {1, 2}; m = 0; t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; While[F[[2]] <= t, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = F[[2]]]; F = {F[[2]], F[[1]] + F[[2]]}]; m];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 09 2025, after Alois P. Heinz *)

PARI

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
a(n) = my(f=factor(fibonacci(n)^2+1)[,1]~, v=select(x->isfib(x), f)); if (#v, vecmax(v), 0); \\ Michel Marcus, Nov 07 2020