On Some Sets with Even Valued Partition Function

Abstract

Let IN be the set of positive integers, ${\cal B}$ = {b₁ < ⋅s < b_h}⊂IN, N∊IN and N≥ b_h. ${\cal A}$ = ${\cal A}$₀(${\cal B}$,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy) such that ${\cal A}${1,..., N} = ${\cal B}$ and p(${\cal A}$,n)≡ 0 (mod 2) for n∊IN and n > N, where p(${\cal A}$,n) denotes the number of partitions of n with parts in ${\cal A}$. Let us denote by σ (${\cal A}$,n) the sum of the divisors of n belonging to ${\cal A}$. In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ(${\cal A}$,2^k n))_{n≥ 1}mod 2^k+1 is periodic with an odd period q_k. In this paper, we will characterize for any fixed odd positive integer q, the sets ${\cal B}$ and the integers N such that q₀ = q, and those for which q_k = q for all k≥ 0. Moreover, a set ${\cal A}$ = ${\cal A}$₀(${\cal B}$,N) is constructed with the property that its period, i.e. the period of (σ(${\cal A}$,n))_{n≥ 1}mod 2, is 217, and for which the counting function is asymptotically equal to that of ${\cal A}$₀({1,2,3,4,5},5) which is a set of period 31.