On the divisor function of sets with even partition functions
For a set \(\mathcal A\) of positive integers let \(p(\mathcal A,n)\) denote the number of partitions of \(n\) into parts from \(\mathcal A\). \textit{J.-L. Nicolas, I. Z. Ruzsa} and \textit{A. Sárközy} [J. Number Theory 73, 292--317 (1998; Zbl 0921.11050)] proved that if \(\emptyset \neq \mathcal B\subseteq \{1,2,\dots,N\}\) then there exists a unique \(\mathcal A\) such that \(\mathcal A\cap\{1,\dots,N\}=\mathcal B\) and \(p(\mathcal A,n)\) is even for \(n>N\). In other words, for a nonconstant polynomial \(P\in \text{GF}(2)[z]\) with \(P(0)=1\), \(\exists!\mathcal A=\mathcal A(P)\) with the property \(\sum_{n\geq 0}p(\mathcal A,n)z^n\equiv P(z)\pmod 2\). Denote by \((\sigma(\mathcal A,n)\) the sum of divisors of \(n\) belonging to \(\mathcal A\). To determine the elements of \(\mathcal A\), it is important to consider the periodicity of the sequences \(S(\mathcal A,k)=(\sigma(\mathcal A,2^kn)\mod{2^{k+1}})_{n\geq 1}\) for all \(k\geq 0\) which was proved by \textit{F. Ben Saïd} and \textit{J.-L. Nicolas} [Acta Arith. 106, 183--196 (2003; Zbl 1052.11069)]. The values of the above sequences were obtained in terms of the Legendre symbol for \(\mathcal A(1+z+z^3)\) (cf. \textit{F. Ben Saïd} and \textit{J.-L. Nicolas} [Sémin. Lothar. Comb. 46, 25 p. (2002; Zbl 1042.11008)]), where as the discrete logarithm was used for \(\mathcal A(1+z+z^3+z^4+z^5)\) cf. \textit{J.-L. Nicolas} [Period. Math. Hung. 43, 177--189 (2001; Zbl 0980.11049)]. In the paper under review the values of \(S(\mathcal A,k)\) are given in terms of orbits under the multiplicative action of \(\langle 2\rangle\leq (Z/\beta Z)^*\) on \(Z/\beta Z\) for odd \(\beta \geq 3\) when \(\text{ord}(P)=2^\alpha\beta\) (the minimal positive integer \(m\) for which \(P(z)\) divides \(1+z^m\) in \(\text{GF}(2)[z]\)).
scientific article; zbMATH DE number 5082275