Lucky numbers are those numbers which survive a sieving process which is similar to the Eratosthenes sieve that can be used to compute the prime numbers.

Let us declare that 1 is a lucky number and let us start with a sieve containing only the odd numbers:

The first number greater than 1 is 3, so we declare 3 lucky and we delete from the sieve all the numbers in a position which is a multiple of 3, i.e., 5, 11, 17, 23, and so on. We are left with The first new survivor is 7, so we declare 7 lucky and we delete from the sieve all the numbers in a position which is a multiple of 7, i.e., 19, 39, and so on. The next lucky number is thus 9 and continuing this ideal process we could identify all the lucky numbers.

The lucky numbers have been studied because they have some properties (like density) which resemble the properties of primes.

The first lucky numbers are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105 more terms

Below, the spiral pattern of lucky numbers up to 3600. See the page on prime numbers for an explanation and links to similar pictures.