Imre Ruzsa observed that since $p/\alpha$ is uniformly distributed equidistributed modulo $1$, we have infinitely many primes $p$ for which the fractional part of $p/\alpha$ is less than $1/\alpha$. Writing $p/\alpha = n_p - {\epsilon}_p$ with $n_p$ an integer and $0 < {\epsilon}_p < 1/\alpha$, we get $p = {\alpha}n_p - {\alpha}\epsilon$ \alpha}{\epsilon}_p$and thus${\lfloor}{\alpha}n_p{\rfloor}$prime for infinitely many distinct positive integers$n_p$. 2 added 1 characters in body Imre Ruzsa observed that since$p/\alpha$is uniformly distributed modulo$1$, we have infinitely many primes$p$for which the fractional part of$p/\alpha$is less than$1/2$. Denoting the 1/\alpha$. Writing $p/\alpha = n_p - {\epsilon}_p$ with $n_p$ an integer part by and $n_p$, 0 < {\epsilon}_p < 1/\alpha$, we get$p = \lfloor\alpha n_p\rfloor${\alpha}n_p - {\alpha}\epsilon$ and thus ${\lfloor}{\alpha}n_p{\rfloor}$ prime for infinitely many distinct positive integers $n_p$.Possibly your asymptotic estimate may be off by a factor of $1/2$, because $p/\alpha$ may round up as well as down.
Imre Ruzsa observed that since $p/\alpha$ is uniformly distributed modulo $1$, we have infinitely many primes $p$ for which the fractional part of $p/\alpha$ is less than $1/2$. Denoting the integer part by $n_p$, we get $p = \lfloor\alpha n_p\rfloor$ for infinitely many distinct positive integers $n_p$. Possibly your asymptotic estimate may be off by a factor of $1/2$, because $p/\alpha$ may round up as well as down.