Strange and pseudo-differentiable functions with applications to prime partitions

Let ${\mathfrak{p}}_{{\mathbb{P}}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of ${\mathfrak{p}}_{{\mathbb{P}}_r}(n)$ as $n\to \mathrm{\infty }$. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.