Lucky Number

Write out all the Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first Odd number $>1$ is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first Odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....

Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (Sloane's A000959). Many asymptotic properties of the Prime Numbers are shared by the lucky numbers. The asymptotic density is $1/\ln N$, just as the Prime Number Theorem, and the frequency of Twin Primes and twin lucky numbers are similar. A version of the Goldbach Conjecture also seems to hold.

It therefore appears that the Sieving process accounts for many properties of the Primes.

See also Goldbach Conjecture, Lucky Number of Euler, Prime Number, Prime Number Theorem, Sieve

References

Gardner, M. ``Mathematical Games: Tests Show whether a Large Number can be Divided by a Number from 2 to 12.'' Sci. Amer. 207, 232, Sep. 1962.

Gardner, M. ``Lucky Numbers and 2187.'' Math. Intell. 19, 26, 1997.

Guy, R. K. ``Lucky Numbers.'' §C3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108-109, 1994.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 100-102, 1988.

Peterson, I. ``MathTrek: Martin Gardner's Luck Number.'' http://www.sciencenews.org/sn_arc97/9_6_97/mathland.htm.

Sloane, N. J. A. Sequence A000959/M2616 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Ulam, S. M. A Collection of Mathematical Problems. New York: Interscience Publishers, p. 120, 1960.

Wells, D. G. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, p. 32, 1986.