Novák-Carmichael numbers and 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 Novák-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 ${\mathcal{N}}_{\mathcal{C}}(x)\gg x{e}^{-(7+o(1))(\log x)\frac{\log \log \log x}{\log \log x}}$, under some reasonable hypothesis.

7 pages