Fonctions de partitions a` parite´ pe´riodique

Let <f>N</f> be the set of positive integers and <f>A</f> a subset of <f>N</f>. For <f>n∈N</f>, let <f>p(A,n)</f> denote the number of partitions of <f>n</f> with parts in <f>A</f>. In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any <f>N∈N</f> and <f>B⊂{1,2,…,N}</f>, there is a unique set <f>A=A0(B,N)</f>, such that <f>p(A,n)</f> is even for <f>n>N</f>. Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) considered <f>σ(A,n)=∑d∣n,d∈Ad</f>, and proved that for all <f>k≥0</f>, the sequence <f>(σ(A,2kn) mod 2k+1)n≥1</f> is periodic on <f>n</f>. In this paper, we generalise the above works for any formal power series <f>f</f> in <f>F2[z]</f> with <f>f(0)=1</f>, by constructing a set <f>A</f> such that the generating function <f>fA</f> of <f>A</f> is congruent to <f>f</f> modulo 2, and by showing that if <f>f=P/Q</f>, where <f>P</f> and <f>Q</f> are in <f>F2[z]</f> with <f>P(0)=Q(0)=1</f>, then for all <f>k≥0</f> the sequence <f>(σ(A,2kn) mod 2k+1)n≥1</f> is periodic on <f>n</f>. [Copyright &y& Elsevier]