On Nov\'ak numbers

TL;DR

This paper establishes new bounds on the count of Novák numbers up to a limit and explores their prime factors, including conditional estimates under the Generalized Riemann Hypothesis.

Contribution

It provides new lower bounds for Novák numbers and, assuming GRH, offers upper estimates for their prime divisors and characterizations of certain Novák-Carmichael numbers.

Findings

New lower bounds for the count of Novák numbers.

Conditional upper bounds on prime divisors of Novák numbers.

Characterization of prime factors of specific Novák-Carmichael numbers.

Abstract

In this work, we obtain some new lower bounds for the number $\mathcal N_B(x)$ of Nov\'ak numbers less than or equal to $x$. We also prove, conditionally on Generalized Riemann Hypothesis, the upper estimates for the number of primes dividing at least one Nov\'ak number and give description for the prime factors of Nov\'ak numbers $N$, such that $2N$ is a Nov\'ak-Carmichael number.