On primes in special sequences with applications to Carmichael numbers

By involving some exponential sums related to $\mathrm{\Lambda }(n)$ in arithmetic progression, we can obtain some new results for von Mangoldt function over {\bf nonhomogeneous} Beatty sequences in arithmetic progressions, which improve some recent results of Banks-Yeager unconditionally. On the other hand, we also considered the primes over Piatetski-Shapiro sequences in arithmetic progressions, which gives a continuous improvement of the results of \cite{BBB}. These results can be used to improve some results related to the Carmichael numbers.