Primes In Fractional Sequences

The results for the fractional sequence $\{[x\mathrm{/}n]+1:n\le x\}$, and the fractional sequence in arithmetic progression $\left {q[x/n]+a:n \leq x\right }$,where$a<q$areintegerssuchthat$\gcd(a,q)=1$, prove that these sequences of fractional numbers contain the set of primes, and the set primes in arithmetic progressions as $x\to \mathrm{\infty }$ respectively. Furthermore, the corresponding error terms for these sequences are improved. Other results considered are the fractional sequences of integers such as the sequence $\left {[x/n]^2+1:n \leq x\right }$generatedbythequadraticpolynomial$n^2+1$, and the sequence $\{[x\mathrm{/}n{]}^3+2:n\le x\}$ generated by the cubic polynomial $n^3+2$. It is shown that each of these sequences of fractional numbers contains infinitely many primes as $x\to \mathrm{\infty }$.