Number theoretic aspects of a combinatorial function

We investigate number theoretic aspects of the integer sequence seq(n) with identification number A000522 in Sloane's On-Line Encyclopedia of Integer Sequences: seq(n) counts the number of sequences without repetition one can build with n distinct objects. By introducing the the notion of the shadow of an integer function, we examine divisibility properties of the combinatorial function seq(n): We show that seq(n) has the reduction property and its shadow d therefore is multiplicative. As a consequence, the shadow d of seq(n) is determined by its values at powers of primes. It turns out that there is a simple characterization of regular prime numbers, i.e. prime numbers p for which the shadow d of seq has the socket property d(p^k)= d(p) for all integers k. Although a stochastic argument supports the conjecture that infinitely many irregular primes exist, there density is so thin that there is only one irregular prime number less than 2500000, namely 383.

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