A "single transit" method of producing a random sequence of integers with a given expected average density is developed, which is simpler, faster, and more flexible than "sieve" devices, and admits of easy statistical analysis, being a straightforward application of the classical "Poisson sequence of trials." Specifically, if F(x) is a suitable function, a sequence P(n), n § 1, may be de¬fined such that, if n is accepted for the random sequence B with probability P(n), then P{ | B(N) - F(N)| < 3P F(N)} > 1 - l/p2(l-p) F(N) - 1, where B(N) is the number of accepted n 5 N in B, and 3P is any prescribed relative error, with 0 < p < 1. Some results on gap distribution, and the "Goldbach property," are included for their mathematical interest. For example, it is shown that almost every sequence of "random primes" contains an infinity of "twins," and for almost every such sequence, every sufficiently large even integer is a sum of two dis¬tinct integers of the sequence.
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