abstract
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Number theoretic aspects of a combinatorial function

Lorenz Halbeisen and Norbert Hungerbühler

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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.