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

TL;DR

This paper explores a variation of the prime divisor function, analyzing its properties and iterative behavior, and provides evidence for conjectures related to its boundedness and pre-image structure.

Contribution

It introduces a new variation of the prime divisor function, studies its properties, and investigates its iterative dynamics, including boundedness and pre-image conjectures.

Findings

No unbounded sequences occur under the new function.

The behavior of the function's iterates resembles aliquot sequences.

Evidence supports analogues of classical conjectures for this variation.

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 $s(n)$.