On a Conjecture of Nicolas–Sárközy about Partitions

Author: Ben saïd F.

Source: Journal of Number Theory, Volume 95, Number 2, August 2002 , pp. 209-226(18)

Publisher: Academic Press

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Abstract:

Let Nopf be the set of positive integers, Bscr ={b₁<…<b_k} sub Nopf , N isin Nopf , and N ges b_k. For I=0 or 1, Ascr = Ascr _I( Bscr ,N) is the set (introduced by Nicolas, Ruzsa, and Sárközy, J. Number Theory 73 (1998), 292–317) such that Ascr cap {1,…,N}= Bscr and p( Ascr ,nm) equiv I(mod2) for n isin Nopf ,n>N, where p( Ascr ,n) denotes the number of partitions of n with parts in Ascr . Let us denote by ( Ascr ,n) the sum of the divisors of n belonging to Ascr . In this paper, we prove that ( Ascr , 2n) mod 4 is periodic with period q₂ multiple of q period of ( Ascr ,n) mod 2; we also give the sets Bscr sub {1,…,5} and the values of N, N les 10, for which q₂q. Moreover, we show that if Ascr (x) is the counting function of Ascr then for Ascr = Ascr ₀({1,2,3},3),lim¯}_xA(x)/x les 1/4. © 2002 Elsevier Science (USA).