The sequences are named after Felix Behrend.
denotes the set of positive integer multiples of members of
The prime numbers form a Behrend sequence, because every integer greater than one is a multiple of a prime number.
of the prime numbers forms a Behrend sequence if and only if the sum of reciprocals of
[1] 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.
[1] The problem of characterizing these sequence was described as "very difficult" by Paul Erdős in 1979.
[2] These sequences were named "Behrend sequences" in 1990 by Richard R. Hall, with a definition using logarithmic density in place of natural density.
[4] Later, Hall and Gérald Tenenbaum used natural density to define Behrend sequences in place of logarithmic density.
[5] This variation in definitions makes no difference in which sequences are Behrend sequences, because the Davenport–Erdős theorem shows that, for sets of multiples, having natural density one and having logarithmic density one are equivalent.