Lucky Numbers

The lucky numbers are obtained by a sieving process that resembles the Sieve of Eratosthenes and the procedure in the Josephus Flavius problem. The lucky numbers sieve has been invented in the 1950 by a group in the Mathematics Division at the Los Alamos National Laboratory, at the time when Stanislaw Ulam was its director.

Start with a sequence of positive integers listed in the natural order. Remove every other number. The first (after 1) left-over number is 3. Mark 3 and remove every third number among the remaining ones. The first left-over number past three is 7. Mark 7 and remove every seventh number among the remaining ones. Continue this way. (The next number is 9.) ...

The numbers that remain after the sieve have been sensibly dubbed lucky. One may observe that 13 belongs to the sequence.

To every one surprise the lucky numbers share some distributive properties with the primes. The fact is startling because the definition of primes depends on their role in the multiplication and products of numbers, whereas the lucky numbers are defined by pure counting. Nonetheless, the distribution of the lucky numbers obeys the same law as that of the primes. The same is true about the twin pairs.

References

1. R. K. Guy, Lucky Numbers, in Unsolved Problems in Number Theory, 2nd ed, Springer-Verlag, pp. 108-109, 1994.
2. C. S. Ogilvy and J. T. Anderson, The Book of Numbers, Oxford University Press, 1966, 102-103
3. N. J. A. Sloane, Sequence A000959 in The On-Line Encyclopedia of Integer Sequences.
4. S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, 1960, p 120
5. S. M. Ulam, Adventures of a Mathematician, University of California Press, 1991
6. D. G. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin, 1986, #32

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