As I was playing around with Mersenne numbers, and discovered the notion of Wagstaff prime going off Wikipedia, I started considering the sequence, for a given $odd$ prime number $p$, defined as follows:
$2^{p}-1, \frac{2^{p}+1}{3}, \frac{2^{p}+3}{5}, \cdots \frac{2^{p}+p}{p+2}$.


It seems that the first term of this sequence is a prime if and only if all other terms but one are positive integers.


Is this true?


Thanks in advance.



Is $2^{p}-1$ prime iff for $\frac{p-1}{2}$ odd positive integers $n$ below $p$, $(n+2)\vert (2^{p}+n)$?
$2^{19}-1$ is prime, but neither $(2^{19}+3)/5$ nor $(2^{19}+9)/11$ is an integer. 


