Divisibility properties of certain recurrent sequences

Let $g$ and $m$ be two positive integers, and let $F$ be a polynomial with integer coefficients. We show that the recurrent sequence $x_0=g$, $x_n=x_{n-1}^n+F(n)$, $n=1,2,3,\dots$, is periodic modulo $m$. Then a special case, with $F(z)=1$ and with $m=p>2$ being a prime number, is considered. We show, for instance, that the sequence $x_0=2$, $x_n=x_{n-1}^n+1$, $n=1,2,3,\dots$, has infinitely many elements divisible by every prime number $p$ which is less than or equal to 211 except for three prime numbers $p=23, 47, 167$ that do not divide $x_n$. These recurrent sequences are related to the construction of transcendental numbers $\zeta$ for which the sequences $[\zeta^{n!}]$, $n=1,2,3,\dots$, have some nice divisibility properties.