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The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base $b$, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base $b$ and omit the proofs for the Thabit numbers of the first kind base $b$ and Cunningham numbers.
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