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A335568 a(n) is the number m such that F(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.

%I A335568 #18 Dec 01 2020 20:54:29
%S A335568 3,3,5,5,7,7,5,7,11,11,13,13,11,13,17,17,5,17,17,7,23,23,7,23,23,5,29,
%T A335568 29,3,29,29,11,7,11,11,7,13,13,0,13,43,43,5,43,47,47,7,47,47,17,7,17,
%U A335568 17,11,3,11,11,3,3,0,7,7,13,13,7,13,23,23,0,23,23,0,5
%N A335568 a(n) is the number m such that F(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.
%C A335568 Fibonacci index of the terms in A338762.
%C A335568 All terms are prime or 0. - _Alois P. Heinz_, Nov 21 2020
%H A335568 Chai Wah Wu, <a href="/A335568/b335568.txt">Table of n, a(n) for n = 1..20000</a>
%F A335568 A000045(a(n)) = A338762(n).
%e A335568 a(10) = 11 because F(10)^2 + 1 = 55^2 + 1 = 3026 = 2*17*89 and 89 = F(11) is the greatest prime Fibonacci divisor of 3026.
%p A335568 a:= proc(n) local i, F, m, t; F, m, t:=
%p A335568       [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
%p A335568       for i from 3 while F[2]<=t do if isprime(F[2]) and
%p A335568         irem(t, F[2])=0 then m:=i fi; F:= [F[2], F[1]+F[2]]
%p A335568       od; m
%p A335568     end:
%p A335568 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 21 2020
%t A335568 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]]}]; m];
%t A335568 Array[a, 100] (* _Jean-François Alcover_, Dec 01 2020, after _Alois P. Heinz_ *)
%Y A335568 Cf. A000040, A000045, A005478, A245306, A338762, A338794 (indices of the 0's).
%K A335568 nonn
%O A335568 1,1
%A A335568 _Chai Wah Wu_, Nov 20 2020