A339082 a(n) is the number m such that F(prime(m)) is the 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, 3, 3, 4, 4, 3, 4, 5, 5, 6, 6, 5, 6, 7, 7, 3, 7, 7, 4, 9, 9, 4, 9, 9, 3, 10, 10, 2, 10, 10, 5, 4, 5, 5, 4, 6, 6, 0, 6, 14, 14, 3, 14, 15, 15, 4, 15, 15, 7, 4, 7, 7, 5, 2, 5, 5, 2, 2, 0, 4, 4, 6, 6, 4, 6, 9, 9, 0, 9, 9, 0, 3, 3, 5, 5, 3, 5, 5, 2, 23, 23, 7

Offset: 1

Author

Chai Wah Wu, Nov 24 2020

Keywords

• nonn

Comments

If a(n) > 0, then prime(a(n)) = A335568(n).

Examples

			a(15) = 7 because F(15)^2 + 1 = 610^2 + 1 = 372101 = 233*1597, 1597 = F(17) is the greatest prime Fibonacci divisor of 372101 and 17 is the 7th prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..30000

Crossrefs

Cf. A000040, A000045, A005478, A245306, A335568, A338762, A338794 (indices of the 0's).

Programs

Maple

a:= proc(n) local i, F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      for i from 3 while F[2]<=t do if isprime(F[2]) and
        irem(t, F[2])=0 then m:=i fi; F:= [F[2], F[1]+F[2]]
      od; numtheory[pi](m)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 25 2020

Mathematica

a[n_] := Module[{i, F = {1, 2}, m = 0, t}, t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; For[i = 3, F[[2]] <= t, i++, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = i]; F = {F[[2]], F[[1]] + F[[2]]}]; PrimePi[m]];
Array[a, 100] (* Jean-François Alcover, Dec 01 2020, after Alois P. Heinz *)

Formula

If A335568(n) = 0, then a(n) = 0, otherwise a(n) = A000720(A335568(n)).