Variants of Korselt's Criterion

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Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer a, there are infinitely many n∈ N such that for each prime factor p|n, we have p-a|n-a. This can be seen as a generalization of Carmichael numbers, which are integers n such that p-1|n-1 for every p|n.