ON THE NUMBER OF DIVISORS OF THE TERMS OF A GEOMETRIC PROGRESSION

Abstract

Let $\left\{a_n\right\}_{n=1}^{\infty}$ be a geometric progression of natural numbers whose quotient has exactly $k$ distinct prime divisors. In this note, we show that the $(k-1)$ th differences of the sequence $\left\{\tau\left(a_n\right)\right\}_{n=1}^{\infty}$ constitute an arithmetic progression. Moreover, we show that there exists a polynomial $p$ of degree $k$ such that $\tau\left(a_n\right)=p(n)$ for each $n \geq 1$.

Received: January 28, 2024
Accepted: February 16, 2024