|
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). 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, ... . 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"〔Gardiner et al (1956)〕 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''. 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 . == See also == *Lucky numbers of Euler *Fortunate number *Happy number *Harshad number 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lucky number」の詳細全文を読む スポンサード リンク
|