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Lucky NumberThis 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,...
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
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