This article is about the formal mathematical concept defined by Stanislaw Ulam; a discussion of the more common meaning is also available.
A lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes. We begin with a list of integers starting with 1:
1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
Then we cross out all even numbers, leaving only the odd integers:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,
The second term in this sequence is 3. Now we cross out every third number which remains in the list:
1, 3, 7, 9, 13, 15, 19, 21, 25,
The third surviving number is now 7 so we cross out every seventh number that remains:
1, 3, 7, 9, 13, 15, 21, 25,
If we repeat this procedure indefinitely, the survivors are the lucky numbers:
- 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ...
Stanislaw Ulam was the first to discuss these numbers, around 1955. He named them "lucky" because of a connection with a story told by the historian Josephus.
Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. It is not known whether there are also infinitely many lucky primes:
- 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, ...
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