On a conjecture of Deaconescu

TL;DR

This paper investigates a number theory conjecture by Deaconescu, proving restrictions on composite numbers that satisfy a specific divisibility condition involving Euler's and Schemmel's totient functions.

Contribution

It establishes that such composite numbers must be odd, squarefree, have at least seven prime factors, and provides an upper bound based on the number of prime divisors.

Findings

Any such n is odd and squarefree.

Such n must have at least seven prime factors.

If n has K prime factors, then n < 2^{2^{K+1}}.

Abstract

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)|\phi(n)-1$, where $\phi(n)$ is Euler's function and $S_2(n)$ is Schemmel's totient function. In this paper we prove that any such $n$ is odd, squarefree and has at least seven distinct prime factors. We also prove that any such $n$ with exactly $K$ distinct prime divisors is necessarily less than $2^{2^{K+1}}$.