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Lucky number

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 which is generated in a similar way as primes are generated from the Sieve of Eratosthenes. We begin with a list of integers starting with 1:

1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
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 again cross out every remaining seventh number:

1,    3,          7,    9,         13,   15,               21,         25,
Finally we get all lucky numbers:


Lucky numbers were named so by Stanislaw Ulam around 1955.

Lucky numbers share some similar properties with primes as their asymptotic behaviour, according to the prime number theorem or the Goldbach's conjecture. There are infinitely many lucky numbers. It is not known whether there are also infinitely many lucky primes:


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Eight (八 pinyin ba1) is considered a lucky number in Chinese culture because it sounds like the word "prosper" (發 pinyin fa1).