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
The first new survivor is 7, so we declare 7 lucky
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.
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many lucky numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p