Radically weakening the Lehmer and Carmichael conditions

TL;DR

This paper explores a weaker condition related to Lehmer's totient problem and Carmichael numbers, showing that the distribution of these numbers is similarly bounded despite their apparent abundance.

Contribution

It introduces a new weaker condition for composite numbers and demonstrates that their distribution is bounded similarly to Carmichael numbers, providing new insights into their rarity.

Findings

The number set satisfying the weakened condition is more numerous than Carmichael numbers.

Their distribution follows a similar upper bound as Carmichael numbers.

The bound is heuristically tight, indicating a fundamental limitation on their density.

Abstract

Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound which is heuristically tight.