Explicit inequalities for the nth lucky number

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 ${\mathrm{\ell }}_n$ denotes the $n$th lucky number then ${\mathrm{\ell }}_n\sim n\log n$, which is analogous to the prime number theorem. This work provides explicit upper and lower bounds on ${\mathrm{\ell }}_n$.