Can every odd number be represented as $2pq-r$ where $p$,$q$, and $r$ are distinct odd primes?

Question

A quick check shows this to be true for odd numbers less than 3289. Is this a known result?

Here is the PARI code I used to check:

x=[]; forprime(p=3, 300, forprime(q=3, 300, forprime(r=3, 300, if(2*p*q-r>0 && p!=q && p!=r && q!=r, x=setunion(x, [2*p*q-r])) )));for(i=1, length(x), if(x[i]<10000,print(x[i])))

The first odd number this particular check fails for is $3289$, but $3289=(2)(5)(347)-181$.

Answer 1

It's very likely that every odd $n$ can be represented this way. In fact, let $p$ be any odd prime that does not divide $n$. Then Dickson's conjecture implies there are infinitely many $q$ such that $q$ and $2pq - n$ are both prime.