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

TL;DR

This paper investigates the congruence properties of the function counting compositions into powers of 2, revealing that it satisfies many modular relations and that v(n) is divisible by high powers of 2 for most n.

Contribution

The paper extends known results on the parity of v(n) to higher powers of 2, providing finite tables for all cases modulo 2^N and showing divisibility properties for large N.

Findings

v(n) satisfies many congruences modulo 2^N

Existence of finite tables listing all cases of v(n) modulo 2^N

v(n) is divisible by 2^N for almost all n

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.