Explicit inequalities for the nth lucky number

Carlo Sanna

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arXiv · 2604.07142

Explicit inequalities for the nth lucky number

Carlo Sanna

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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 $\ell_n$.

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BibTeX RIS

Carlo Sanna. 2026-04-08. Explicit inequalities for the nth lucky number. https://arxiv.org/abs/2604.07142

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