\(l\)-divisibility of \(l\)-regular partition functions
For \(l>1\) a partition of \(n\) is called \(l\)-regular if it has no summand divisible by \(l\). Let \(b_l(n)\) be the number of \(l\)-regular partitions of \(n\). The authors give exact criteria for the divisibility of \(b_l(n)\) by \(l\) for \(l\in\{5,7,11\}\). For example, in the case \(l=5\) let \[ 6n+1=\prod_{i=1}^sp_i^{e_i} \] be the prime factorization of \(6n+1\) (\(p_1,\ldots,p_s\) are distinct primes and \(e_i>0\) for all \(1\leq i\leq s\)). For each \(i\) with \(p_i\equiv 1\,(\bmod\,3)\) write \(p_i=x_i^2+3y_i^2\) with \(x_i,y_i\in\mathbb{Z}\). \textbf{Theorem.} Let \(n\geq 1\). Then \(b_5(n)\) is divisible by \(5\) if and only if for some \(1\leq i\leq s\) one of the following holds: (1) \(p_i=5.\) (2) \(p_i\equiv 2\,(\bmod\,3)\), \(p_i\neq 5\) and \(e_i\) is odd. (3) \(p_i\equiv 1\,(\bmod\,3)\), \(5|x_i\) and \(e_i\) is odd. (4) \(p_i\equiv 1\,(\bmod\,3)\), \(5|y_i\) and \(e_i\equiv 4\,(\bmod\,5)\). (5) \(p_i\equiv 1\,(\bmod\,3)\), \(5|(x_i^2-y_i^2)\) and \(e_i\equiv 2\,(\bmod\,3)\). (6) \(p_i\equiv 1\,(\bmod\,3)\), \(5\nmid x_iy_i(x_i^2-y_i^2)\) and \(e_i\equiv 5\,(\bmod\,6)\).
scientific article; zbMATH DE number 5636940