Shifts of the prime divisor function of Alladi and Erd\H{o}s

Abstract

We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erd\H{o}s, a close relative of the sum of proper divisors function $s(n)$. After proving some basic properties regarding these functions, we study the dynamics of its iterates and discover behaviour that is reminiscent of aliquot sequences. We prove that no unbounded sequences occur, analogous to the Catalan-Dickson conjecture, and give evidence towards the analogue of the Erd\H{o}s-Granville-Pomerance-Spiro conjecture on the pre-image of $s(n)$.