Lucky number

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:

1,3,7,9,13,15,21,25,31,33,37,43,49,51,63,67,69,73,75,79,87,93,99,...

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:

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
• 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