Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.3 |

Institute of Mathematics, Department of Integrative Biology

Universität für Bodenkultur

Gregor Mendel-Straße 33

A-1180 Wien

Austria

and

Vilnius University

Department of Mathematics and Informatics

Naugarduko 24

LT-03225 Vilnius

Lithuania

**Abstract:**

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

(Concerned with sequences A000120 A000123 A018819 A023359.)

Received March 4 2010;
revised version received April 29 2010.
Published in *Journal of Integer Sequences*, May 3 2010.

Return to