On a Conjecture of Nicolas–Sárközy about Partitions Original Research Article

Let image be the set of positive integers, image={b1<…N, where p(image,n) denotes the number of partitions of n with parts in image. Let us denote by σ(image,n) the sum of the divisors of n belonging to image. In this paper, we prove that σ(image, 2n) mod 4 is periodic with period q2 multiple of q period of σ(image,n) mod 2; we also give the sets imagesubset of{1,…,5} and the values of N, Nless-than-or-equals, slant10, for which q2≠q. Moreover, we show that if image(x) is the counting function of image then for image=image0({1,2,3},3),image}x→∞A(x)/xless-than-or-equals, slant1/4.