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Lucky number

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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, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

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 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,...

External links


sl:srečno število

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