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Number Theoretic Aspects of a Combinatorial Function
"... We investigate number theoretic aspects of the integer sequence seq 1 1 (n) with identication number A000522 in Sloane's OnLine Encyclopedia of Integer Sequences: seq 1 1 (n) counts the number of sequences without repetition one can build with n distinct objects. By introducing the the not ..."
Abstract

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We investigate number theoretic aspects of the integer sequence seq 1 1 (n) with identication number A000522 in Sloane's OnLine Encyclopedia of Integer Sequences: seq 1 1 (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 1 1 (n): We show that seq 1 1 (n) has the reduction property and its shadow d therefore is multiplicative. As a consequence, the shadow d of seq 1 1 (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 1 has the socket property d(p k ) = d(p) for all integers k. Although a stochastic argument supports the conjecture that innitely many irregular primes exist, there density is so thin that there is only one irregular prime number less than 2:5 ...