Multiplicity one prime in the factorisation of p-N   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.
 A: The number of primes in $[N,3N/2]$ grows as $\frac{N}{\log N}$, while the number of powerful numbers 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.
A: I cannot think of an exact reference but the result you are looking for can be obtained as follows:
The number of primes $p\in(N,2N]$ such that $p-N$ is divisible by the square of a prime $q>\log N$ is $\ll N/\log^2N$ (this follows by any upper bound sieve). Also, the number of primes $p\in(N,2N]$ such that $p-N$ is composed only of prime numbers $\le\log N$ is at most the number of integers $m\le N$ which are $\log N$-smooth (i.e. have only only prime factors $\le\log N$). The number of such integers is at most $N^{1-1/2\log\log N}\ll N/\log^2N$ (see for example Theorem 1, Chapter III.5, in Tenenbaum's book "Introduction to analytic and probabilistic number theory"). So 
$$|\{N < p\le2N:\exists q~{\rm prime}~{\rm with}~q>\log N~{\rm and}~q\|p-N\}|=\frac N{\log N}+O\Bigl(\frac N{\log^2N}\Bigr).$$
