Abstract:

Let $\vartheta (n)$ denote the number of compositions (ordered partitions) of a positive integer $n$ into powers of $2$. It appears that the function $\vartheta (n)$ satisfies many congruences modulo $2^{N}$. For example, for every integer $a$ there exists (as $k$ tends to infinity) the limit of $\vartheta (2^k+a)$ in the $2-$adic topology. The parity of $\vartheta (n)$ 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 $\vartheta (n)$ is divisible by $2^{N}$ for almost all $n$, however large the value of $N$ is.