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

On primes in special sequences with applications to Carmichael numbers