Let $F_n$ be the $n$ th fibonacci-number and define $$f(n):=F_{n^2}+F_n+1$$
For which positive integers $n$ do we have no small prime factor (say $p<10^7$) $p\mid f(n)$ ? Are there useful criterions for those $n$ ?
$f(n)$ is prime for $n=1,2,3,4,10$ and no other $n\le 700$
We have no prime factor below $10^7$ for the following positive integers ($10$ is excluded) : $[21, 26, 27, 32, 50, 51, 52, 60, 66, 92, 93, 98, 106, 120, 132, 146, 171, 180, 204, 218, 219, 220, 258, 290, 291, 292, 309, 412, 424, 429, 450, 460, 464, 472, 492, 500, 506, 532, 540, 544, 549, 552, 570, 584, 610, 650, 651, 660, 666, 693, 699, 700]$
The first case where factordb knows no prime factor is $n=92$.