Lucky Number

In number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes.

Begin with a list of integers starting with 1:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,

Every second number (all even numbers) is eliminated, leaving only the odd integers:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,

The second term in this sequence is 3. Every third number which remains in the list is eliminated:

1, 3, 7, 9, 13, 15, 19, 21, 25,

The next surviving number is now 7, so every seventh number that remains is eliminated:

1, 3, 7, 9, 13, 15, 21, 25,

When this procedure has been carried out completely, the survivors are the lucky numbers:

1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ... (sequence A000959 in OEIS).

The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve, "the sieve of Josephus Flavius" 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. 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.

Because of these apparent connections with the prime numbers, some mathematicians have suggested that these properties may be found in a larger class of sets of numbers generated by sieves of a certain unknown form, although there is little theoretical basis for this conjecture. Twin lucky numbers and twin primes also appear to occur with similar frequency.

A lucky prime is a lucky number that is prime. It is not known whether there are infinitely many lucky primes. The first few are

3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193 (sequence A031157 in OEIS).