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 1 has the socket property d(pk) = 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 2.5 · 106, namely 383.