lucky number

A lucky number is an integer that remains after a sieving process similar to a sieving process for prime numbers. The first few 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, etc., listed in A000959 of Sloane’s OEIS. There are infinitely many lucky numbers. These numbers share some properties with prime numbers, mostly in regards to distribution. Stanisław Ulam (http://planetmath.org/StanislawUlam) was the first to study these numbers.

The sieve process for the lucky numbers begins with a list of odd positive integers from 1 to whatever limit one wishes (let’s say 49). We circle 1 and 3 and cross out every third number that remains (counting from the beginning):

$$1,3,\mathrm{5̸},7,9,\cancel{11},13,15,\cancel{17},19,\mathrm{\dots }$$

Then we circle the number next to the one that we last circled and cross out every $x$th term as indicated by the number we just circled, in this case, 7, starting the count from the beginning but not counting numbers that have already been struck out:

$$1,3,\mathrm{5̸},7,9,\cancel{11},13,15,\cancel{17},\cancel{19},\mathrm{\dots }$$

This step is repeated until every number in our list has been either circled or crossed out. The numbers that remain are “lucky” because they survived the process without ever being struck.

The “lucky number theorem” is almost the same as the prime number theorem.