On a conjecture of Nicolas-Sárközy about partitions
Let \(\mathcal B\) be a non-empty set of distinct positive integers with largest element \(b\), and let \(\mathcal A=\mathcal A_i(\mathcal B,N)\) for \(N\geq b\) be the unique set such that \(\mathcal A\cap \{1,2,\dots,N\}=\mathcal B\) and \[ p(\mathcal A,n)\equiv i\pmod 2 \quad {\text{ for }} n\in\mathbb N,\;n>N. \] Here \(p(\mathcal A,n)\) is the number of partitions of \(n\) with parts in \(\mathcal A\). The set \(\mathcal A_i(\mathcal B,N)\) was first introduced by \textit{J.-L. Nicolas}, \textit{I. Z. Ruzsa} and \textit{A. Sárközy} [J. Number Theory 73, 292--317 (1998; Zbl 0921.11050)] and for example, for \(N\geq 2\), \[ \begin{aligned} \mathcal A_0(\{1\},N-1)&=\{1,N,2N,2^2N,2^3N,\dots\},\\ \mathcal A_1(\{1\},N-1)&=\{1\}.\end{aligned} \tag{\(*\)} \] Given the set \(\mathcal A\), let \(\sigma(\mathcal A,n)\) be the divisor function \[ \sigma(\mathcal A,n)=\sum_{ d| n,\;d\in\mathcal A} d. \] Then \(\sigma(\mathcal A,n)\pmod 2\) is periodic in the variable \(n\) with period \(q\). If \(q_2\) is the period of \(\sigma(\mathcal A,2n)\pmod 4\) then the main result of the paper is that \(q_2\) is a multiple of \(q\). For example, for the set \(\mathcal A\) given in (\(*\)), \(q=q_2=N\) if \(N\) is odd, \(q=1\) if \(N\) is even, \(q_2=N/2\) if \(N\equiv 2\pmod{4}\) and \(q_2=1\) if \(N\equiv 0\pmod{4}\). The above result concerning \(q\) and \(q_2\) implies a conjecture by Nicolas and Sárközy stating that if \(\mathcal A=\mathcal A_0(\{1,2,3\},3)\) and \(n\) is even then \[ \sigma(\mathcal A,n)\equiv 1,2,3 \pmod 4 \quad {\text{ as }} \Bigl(\frac{n}{7}\Bigr)=0,-1,1, \] respectively, where \(\bigl(\frac{n}{7}\bigr)\) is the Legendre symbol.
scientific article; zbMATH DE number 1815502