Arithmetic properties of the sequence of derangements

The sequence of derangements is given by the formula D0=1D0=1, Dn=nDn−1+(−1)nDn=nDn−1+(−1)n, n>0n>0. It is a classical object appearing in combinatorics and number theory. In this paper we consider such arithmetic properties of the sequence of derangements as: periodicity modulo d  , where d∈N+d∈N+, p-adic valuations and prime divisors. Next, we use them to establish arithmetic properties of the sequences of even and odd derangements.