A congruence for \(c{\varphi{}}_{h,k}(n)\)
Let \(c\varphi_{h,k}(n)\) be the number of generalized Frobenius partitions of \(n\) with \(h\) colors and at most \(k\) repetitions, as introduced by \textit{Padmavathamma} [Fibonacci Q. 27, 125-130 (1989; Zbl 0669.10025)]. It is proved that \[ \sum_{d\mid(h,n)} \mu(d) c\varphi_{h/d,k}(n/d)\equiv 0\pmod{hH}, \] where \(H\) is the product of all the prime power factors of \(h\) which are relatively prime to \(k+1\). As an immediate corollary, \[ \sum_{d\mid(h,n)}\mu(d)c\varphi_{h/d,k}(n/d)\equiv 0\pmod h^ 2 \] if \((h,k+1)=1\).
scientific article; zbMATH DE number 168037