There are several types of numbers that are commonly termed "lucky numbers."
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
asymptotic density is ,
just as the prime number theorem, and the
frequency of twin primes and twin lucky numbers are
similar. A version of the Goldbach conjecture
also seems to hold.
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, ....
,
just as the prime number theorem, and the
frequency of twin primes and twin lucky numbers are
similar. A version of the Goldbach conjecture
also seems to hold.
