Lucky number

In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers).

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
Every second number (all even numbers) in the remaining list is eliminated, 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. Starting with 5, every third number which remains in the list is eliminated:
1		3				7		9				13		15				19		21				25
The next surviving number is now 7. Every seventh remaining number is eliminated, starting with 19:
1		3				7		9				13		15						21				25

One way that the application of the procedure differs to that of the Sieve of Eratosthenes is, for n being the number being multiplied on a specific pass, the first number eliminated on the pass is the n-th remaining number that hasn't yet been eliminated, as opposed to the number 2n. That is to say that the numbers this sieve counts through is different on each pass (for example 1, 3, 7, 9, 13, 15, 19... on the third pass), whereas in the Sieve of Eratosthenes, the sieve always counts through the entire original list (1, 2, 3...).

When this procedure has been carried out completely, 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).

The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve, "the sieve of Josephus Flavius"^[1] because of its similarity with the counting-out game in the Josephus problem.

Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. However, if L_n denotes the n-th lucky number, and p_n the n-th prime, then L_n > p_n for all sufficiently large n.^[2]

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).

See also

• Lucky numbers of Euler
• Fortunate number
• Happy number
• Harshad number

References

1. Gardiner et al (1956)
2. Hawkins, D.; Briggs, W.E. (1957). "The lucky number theorem". Mathematics Magazine 31 (2): 81–84,277–280. doi:10.2307/3029213. ISSN 0025-570X. Zbl 0084.04202.

• Gardiner, Verna; Lazarus, R.; Metropolis, N.; Ulam, S. (1956). "On certain sequences of integers defined by sieves". Mathematics Magazine 29 (3): 117–122. doi:10.2307/3029719. ISSN 0025-570X. Zbl 0071.27002.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. C3. ISBN 978-0-387-20860-2. Zbl 1058.11001.

External links

• Peterson, Ivars. MathTrek: Martin Gardner's Lucky Number
• Weisstein, Eric W., "Lucky Number", MathWorld.
• Lucky Numbers by Enrique Zeleny, The Wolfram Demonstrations Project.
• Symonds, Ria. "31: And other lucky numbers". Numberphile. Brady Haran.