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lucky number





A number in a sequence, first identified and named around 1955 by Stanislaw Ulam, that evades a particular type of number "sieve" (similar to the famous Sieve of Eratosthenes), which works as follows. Start with a list of integers, including 1, and cross out every second number: 2, 4, 6, 8, ... The second surviving integer is 3. Cross out every third number not yet eliminated. This removes 5, 11, 17, 23, ... The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Repeat this process indefinitely and the numbers that survive are the "lucky" ones:
1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195 ...
Amazingly, though generated by a sieve based entirely on a number's position in an ordered list, the luckies share many properties with prime numbers. For example, there are 25 primes less than 100 and 23 luckies less than 100. In fact, it turns out that primes and luckies crop up about equally often within given ranges of integers. Also, the gaps between successive primes and the gaps between successive luckies widen at roughly the same rate as the numbers increase, and the number of twin primes – primes that differ by 2 – is close to the number of twin luckies. The luckies even have their own equivalent of the famous (still unsolved) Goldbach Conjecture, which states that every even number greater than 2 is the sum of two primes. In the case of luckies, it is conjectured that every even number is the sum of two luckies; no exception has yet been found. Another unresolved problem is whether there are an infinite number of lucky primes.


Related entry

   • Ulam Spiral


Related category

   • TYPES OF NUMBERS