Lucky Number

There are several types of numbers that are commonly termed "lucky numbers."

The first is the lucky numbers of Euler. The second is obtained by writing out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number  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, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is , 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.

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." https://www.sciencenews.org/sn_arc97/9_6_97/mathland.htm.

Sloane, N. J. A. Sequence A000959/M2616 in "The On-Line Encyclopedia of Integer Sequences."

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.