Repeated values of some restricted divisor functions

TL;DR

This paper proves that for any positive integers k and B, the restricted divisor function d_k(n) equals d_k(n+B) infinitely often, revealing a new infinite recurrence property of these functions.

Contribution

It establishes the infinite occurrence of equal values of the restricted divisor function shifted by B, a novel result in divisor function theory.

Findings

d_k(n)=d_k(n+B) occurs infinitely often

The result applies to any positive integers k and B

Provides new insights into divisor function behavior

Abstract

We prove that $d_k(n)=d_k(n+B)$ infinitely often for any positive integers $k$ and $B$, where $d_k(n)$ denotes the number of divisors of $n$ coprime to $k$.