Lucky number

In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers).^[1]

The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. In the same work they also suggested calling another sieve, "the sieve of Josephus Flavius"^[2] because of its similarity with the counting-out game in the Josephus problem.

Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and twin primes also appear to occur with similar frequency. However, if L_n denotes the n-th lucky number, and p_n the n-th prime, then L_n > p_n for all sufficiently large n.^[3]

Because of their apparent similarities with the prime numbers, some mathematicians have suggested that some of their common properties may also be found in other sets of numbers generated by sieves of a certain unknown form, but there is little theoretical basis for this conjecture.