On Certain Sequences of Integers Defined by Sieves

Abstract

The sequence of primes can be defined by the sieve of Eratosthenes. One can think of variations in the definition of a sieve. The following problem was studied by means of actual calculation on an electronic computing machine (partly with the aim of trying to develop coding procedures which would obviate the need for large memories which at first appear necessary for problems of this sort). Consider the sequence of all positive integers, 1, 2, 3, e,. . We shall now strike out from this sequence every second term by counting from 1. The odd integers will be left. We shall now strike out every third integer in the remaining sequence, again starting to count from 1, but considering only the rema-ini.ng integers. We shall obtain a second sequence of integers. The next step is to strike out every fourth integer counting only the remaining ones and we obtain another subsequence. We can continue this process Indefinitely. It is obvious that infinitely many integers will remain after we have completed the process. This sieve is different from that of Eratosthenes since in striking out all multiples of successive integers we count off only among the remaining ones. It could perhaps be called a sieve of Josephus Flavius.The result of this sieve is a sequence of integers of a density much smaller than that of the primes. Another sieve could be the following: consider again the sequence of integers starting with 1. We shall strike out from it every second term.Apart from 1, the first integer which remains is 3; now in the remaining sequence we shall strike off every term whose index is a multiple of 3. In the sequence which remains now and which consists of 1, 3, 7, 9, 13, 15, 19, ..., the first term which has not been used already is 7. We shall,therefore, strike off every term in this sequence whose index is a multiple of 7; that is to say, the 7th (which is 19),, 14th, 21st, etc., term in this sequence. In the remaining sequence we shall look up the first term which has not been used (it is 9) and again strike off terms whose indices are multiples of it. We continue this process indefinitely; it is obvious again that infinitely many terms remain. They may be called the result of our sieve. The aim of our exercise was to consider certain asymptotic properties of this latter sequence of numbers, let us call them for brevity numbers. All lucky numbers up to 48,000 were quickly computed on the machine and the following data about them were obtained.(For the first few see Table I.)