THE RAMANUJAN JOURNAL, Vol. 10, No. 3 (2005), 383-394.

# Arithmetic Properties of Summands of Partitions II

CÉCILE DARTYGE
Institut Élie Cartan, Université Henri Poincaré--Nancy 1, BP 239, 54506 Vand\oe uvre Cedex, France
Email: dartyge@iecn.u-nancy.fr

ANDRÁS SÁRKÖZY
Department of Algebra and Number Theory, Eötvös Loránd University, H-1518 Budapest, Pf. 120, 1117 Budapest, Pázmány Péter Sétány 1/C, Hungary
Email: sarkozy@cs.elte.hu

Received December 27, 2002; Accepted April 7, 2003

Abstract. Let $d\in\N$, $d\ge 2$. We prove that a positive proportion of partitions of an integer $n$ satisfies the following: for all $1\le a < b \le d$, the number of the parts congruent to $a \pmod{d}$ is greater than the number of the parts congruent to $b \pmod{d}$. We also show that for almost all partitions the rate of the number of square free parts is $\frac{6}{\pi^2}(1 + o(1))$.

Keywords: partitions, residue classes

2000 Mathematics Classification: Primary 11P82

MATHSCINET: 2193385