Lucky number

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This article is about the formal mathematical concept. For other uses, see Lucky number (disambiguation).

In number theory, 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:

1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ... (sequence A000959 in OEIS).

Image:LuckySieve.gif

An animation demonstrating the lucky number sieve. The numbers in red are 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. Because of these apparent connections with the prime numbers, some mathematicians have suggested that these properties may be found in a larger class of sets of numbers generated by sieves of a certain unknown form, although there is little theoretical basis for this conjecture. Twin lucky numbers and twin primes also appear to occur with similar frequency.

A lucky prime is a lucky number that is prime. It is not known whether there are infinitely many lucky primes. The first few are

3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193 (sequence A031157 in OEIS).