Let $a_n$ be an increasingsequence of positive reals with $a_n \rightarrow \infty$ as $n \rightarrow \infty$.Necessary andsufficient conditions are obtained for each of the sequences $[\alpha a_n], [\alpha^{a_n}],[{a_n}^{\alpha}]$ to take on infinitely many prime values for almost all $\alpha > \beta$.Forexample, the sequence $[\alpha a_n]$ is infinitely often prime for almost all $\alpha > 0$ if and onlyif there is a subsequence of the $a_n$, say $b_n$, with $b_{n+1} > b_n + 1$and with the series $\sum1/{b_n}$ divergent.Asymptotic formulae are obtained when the sequences considered are lacunary.Anearlier result of the author reduces the problem to estimating the measure of overlaps of certainsets, and sieve methods are used to obtain the correct order upper bounds.

1991 MathematicsSubject Classification: primary 11N05; secondary 11K99, 11N36.