Even partition functions
For a set \(A\) of positive integers, \(p(A,n)\) denotes the number of partitions of \(n\) into parts from \(A\). Given an integer \(N\) and a set \(B \subset \{1, 2, .., N\}\) one easily sees that there is a unique set \(A\) such that \(A \bigcap [1,N] = B\) and \(p(A,n)\) is even for \(n>N\). Initially it was believed that such sets are completely irregular. This paper shows that, on the contrary for \(N=3\) and \(B=\{1,2,3\}\) the corresponding set \(A\) has a rich multiplicative structure. Elements of \(A\) are composed exclusively of primes congruent to 3, 5 or 6 modulo 7 (except 2 and 7), and within such numbers there are further conditions to determine membership in \(A\). Based on this, the authors are able to prove that the cardinality of \(A\) up to \(x \) is asymptotic to \( c x ( \log x)^{-3/4}\).
scientific article; zbMATH DE number 1889085