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\begin{document}

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

\end{document}

                                                                                

