Behrend sequence
In number theory, a Behrend sequence is an integer sequence whose multiples include almost all integers. The sequences are named after Felix Behrend. Definition. If formula_1 is a sequence of integers greater than one, and if formula_2 denotes the set of positive integer multiples of members of formula_1, then formula_1 is a Behrend sequence if formula_2 has natural density one. This means that the proportion of the integers from 1 to formula_6 that belong to formula_2 converges, in the limit of large formula_6, to one. Examples. The prime numbers form a Behrend sequence, because every integer greater than  one is a multiple of a prime number. More generally, a subsequence formula_1 of the prime numbers forms a Behrend sequence if and only if the sum of reciprocals of formula_1 diverges. The semiprimes, the products of two prime numbers, also form a Behrend sequence. The only integers that are not multiples of a semiprime are the prime powers. But as the prime powers have density zero, their complement, the multiples of the semiprimes, have density one. History. The problem of characterizing these sequence was described as "very difficult" by Paul Erdős in 1979. These sequences were named "Behrend sequences