A lucky number is a number belonging to one of several classes commonly described as "lucky."
The first is the lucky numbers of Euler. The second is obtained by writing out all odd numbers:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd
number
is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, ....
The first odd number greater than 3 in the list is
7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....
Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS
A000959). Many asymptotic properties of the
prime numbers are shared by the lucky numbers. The
number of lucky numbers not exceeding is asymptotic to , so their natural density
is 0. This counting asymptotic is analogous to the prime
number theorem (Hawkins and Briggs 1958). Numerical observations also suggest
similarities between the frequencies of twin primes
and twin lucky numbers. A version of the Goldbach
conjecture for lucky numbers remains conjectural.
It therefore appears that the sieving process accounts
for many properties of the primes.
is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, ....
The first odd number greater than 3 in the list is
7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....
is asymptotic to 
, so their natural density
is 0. This counting asymptotic is analogous to the prime
number theorem (Hawkins and Briggs 1958). Numerical observations also suggest
similarities between the frequencies of twin primes
and twin lucky numbers. A version of the Goldbach
conjecture for lucky numbers remains conjectural.
