Explicit inequalities for the nth lucky number

TL;DR

This paper establishes explicit upper and lower bounds for the nth lucky number, a sequence generated by a sieve similar to that for primes, highlighting its distributional similarities to prime numbers.

Contribution

It provides the first explicit inequalities for the nth lucky number, enhancing understanding of its growth and distribution.

Findings

Derived explicit bounds for lucky numbers

Confirmed the asymptotic behavior ll_n  n \u2217 g n

Linked lucky number distribution to prime number theorem

Abstract

Gardiner, Lazarus, Metropolis, and Ulam introduced a variation of the sieve of Eratosthenes that (instead of producing the sequence of prime numbers) produces the sequence of "lucky numbers". The distribution of lucky numbers has a striking similarity to that of prime numbers. In particular, Hawkins and Briggs proved that if $\ell_n$ denotes the $n$th lucky number then $\ell_n \sim n \log n$, which is analogous to the prime number theorem. This work provides explicit upper and lower bounds on $\ell_n$.