# Nov\'ak-Carmichael numbers and shifted primes without large prime factors

## TL;DR

This paper establishes new lower bounds for counting Novák-Carmichael numbers, linking their distribution to shifted primes without large prime factors, and provides both unconditional and conditional estimates.

## Contribution

It introduces novel lower bounds for Novák-Carmichael numbers' counting function based on properties of shifted primes without large prime factors.

## Key findings

- Unconditional lower bound: x^{0.7039-o(1)}
- Conditional lower bound involving exponential decay
- Results connect Novák-Carmichael numbers to shifted primes without large prime factors

## Abstract

We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Nov\'ak-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that $\mathcal N_{\mathcal C}(x) \gg x^{0.7039-o(1)}$ unconditionally and that $\mathcal N_{\mathcal C}(x) \gg xe^{-(7+o(1))(\log x)\frac{\log\log\log x}{\log\log x}}$, under some reasonable hypothesis.