An odd prime is called a cluster prime if every even
positive integer less than can be written as a difference of two primes , where . The first 23 odd primes 3, 5, 7, ..., 89 are all
cluster primes. The first few odd primes that are not cluster primes are 97, 127,
149, 191, 211, ... (OEIS A038133).

The numbers of cluster primes less than , , ... are 23, 99, 420, 1807, ... (OEIS A039506),
and the corresponding numbers of noncluster primes are 0, 1, 68, 808, 7784, ... (OEIS
A039507). It is not known if there are infinitely
many cluster primes, but Blecksmith et al. (1999) show that for every positive
integer ,
there is a bound
such that if ,
then

where
is the number of cluster primes not exceeding . Blecksmith et al. (1999) also show that the sum of
the reciprocals of the cluster primes is finite.

Blecksmith, R.; Erdős, P.; and Selfridge, J. L. "Cluster Primes." Amer. Math. Monthly106, 43-48, 1999.Elsholtz,
C. "On Cluster Primes." Acta Arith.109, 281-284, 2003.Sloane,
N. J. A. Sequences A038133, A039506,
and A039507 in "The On-Line Encyclopedia
of Integer Sequences."