# Lucky number

*This article is about the formal mathematical concept. See Numerology for a discussion of the more common meaning.*

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 every second number (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:

Stanisław 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

- Peterson, Ivars:
*MathTrek: Martin Gardner's Lucky Number*: http://www.sciencenews.org/sn_arc97/9_6_97/mathland.htm - Schneider, Walter:
*A list of the first 1000 lucky numbers*: http://www.wschnei.de/number-theory/lucky-numbers-list.html - Sloane, Neil J. A.:
*A sequence of lucky numbers - A000959*: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000959 - Sloane, Neil J. A.:
*A sequence of lucky primes - A031157*: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=031157 - Sloane, Neil J. A.:
*A sequence of composite lucky numbers - A031879*: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=031879 - Weisstein, Eric W.:
*Lucky Number*: http://mathworld.wolfram.com/LuckyNumber.html

de:Glückliche Zahl fr:Nombre chanceux he:מספר מזל (מתמטיקה) nl:Geluksgetal sl:Srečno število