Congruence properties of the function which counts compositions into powers of 2

Abstract

Let v(n) denote the number of compositions (ordered partitions) of a positive integer n into powers of 2. It appears that the function v(n) satisfies many congruences modulo 2^N. For example, for every integer B there exists (as k tends to infinity) the limit of v(2^k+B) in the 2-adic topology. The parity of v(n) obeys a simple rule. In this paper we extend this result to higher powers of 2. In particular, we prove that for each positive integer N there exists a finite table which lists all the possible cases of this sequence modulo 2^N. One of our main results claims that v(n) is divisible by 2^N for almost all n, however large the value of N is.