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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.

%I A339082 #21 Jul 15 2021 18:54:53
%S A339082 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,
%T A339082 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,
%U A339082 6,4,6,9,9,0,9,9,0,3,3,5,5,3,5,5,2,23,23,7
%N 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.
%C A339082 If a(n) > 0, then prime(a(n)) = A335568(n).
%H A339082 Chai Wah Wu, <a href="/A339082/b339082.txt">Table of n, a(n) for n = 1..30000</a>
%F A339082 If A335568(n) = 0, then a(n) = 0, otherwise a(n) = A000720(A335568(n)).
%e A339082 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.
%p A339082 a:= proc(n) local i, F, m, t; F, m, t:=
%p A339082       [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
%p A339082       for i from 3 while F[2]<=t do if isprime(F[2]) and
%p A339082         irem(t, F[2])=0 then m:=i fi; F:= [F[2], F[1]+F[2]]
%p A339082       od; numtheory[pi](m)
%p A339082     end:
%p A339082 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 25 2020
%t A339082 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]];
%t A339082 Array[a, 100] (* _Jean-François Alcover_, Dec 01 2020, after _Alois P. Heinz_ *)
%Y A339082 Cf. A000040, A000045, A005478, A245306, A335568, A338762, A338794 (indices of the 0's).
%K A339082 nonn
%O A339082 1,1
%A A339082 _Chai Wah Wu_, Nov 24 2020