if I wanna know how many prime number $p$ less than $x$, such that :
there is at least one prime number of this form$ 2Kp+1$ where $K={1,2,3,4,5,6,7,8,9,10}$.
for example:
if $x=1000$, then the exact value of the primes of the form $2Kp+1$, where $p>x$, and $K=1,2,.....,10$ is $395$, which very easy to predict by using hardy littlewood conjecture, and in this case it will be $$2C \frac{1000}{\ln^2 1000} \sum_{K=1}^{10} \prod_{2<q|2K} \frac{q-1}{q-2}=384$$ .
now I don't wanna predict this $395$, I wanna predict the number of the primes less than $1000$ such that:
there is at least one prime number of this form$ 2Kp+1$ where $K={1,2,3,4,5,6,7,8,9,10}$. which in this case equals to $162$
is there any estimate that predicts how many they are ?