I'm wondering if analytic number theorists can prove results which have the following flavor:


So let $N$ be a large positive integer.


Q: Can you always find a prime number $p$ in the interval $(N, 3N/2)$ for which
there exists an odd prime $q$ which divides $p-N$ with multiplicity exactly one?


If such a result can be found in the literature I would like to have a reference. 
I have just not the single idea about where to start in order to prove such a result. 


I kind of remember vaguely that every large enough even integer $N$ can be written as 
$p\_1+p\_2p\_3$ where the $p\_i$'s are prime numbers which is not that far from what I'm asking for.



Multiplicity one prime in the factorisation of p-N  
The number of primes in $[N,3N/2]$ grows as $\frac{N}{\log N}$, while the number of [powerful numbers](http://en.wikipedia.org/wiki/Powerful_number) in $[1,N/2]$ grows as $\sqrt{N}$, so pretty quickly you will find primes $p\in [N,3N/2]$ so that $p-N$ is not powerful, i.e. has a prime divisor which has multiplicity 1.


