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Lucky numbers

In Matematica on settembre 8, 2009 at 4:28 pm

Lucky numbers have this name because they have the luck to survive to a selective process of elimination. The elimination’s method is analogue to the Sieve of Eratosthenes. Let us begin eliminating every even numbers, obtaining the sequence of odd numbers:

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31…

The first number that survive after 1 is 3. Let us proceed eliminating every third number from the last sequence:

1 3 7 9 13 15 19 21 25 27 31…

The number that survive after 3 is 7. Then let us proceed eliminating from the last sequence every 7th number and so on… The survived numbers form the sequence of lucky numbers:

1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75…

These numbers have similar property to prime numbers. For example their asymptotic density is the same of prime numbers. So it’s possible that the asymptotic property of prime numbers don’t derive from their definition (a number is prime if it is divisible only for one and itself), but because they are obtained through an operation of “Sieve”. So every sequence, obtained by these processes, can have the same asymptotic propery.

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