A cluster prime is an odd prime 
such that 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 3, 23, 99, 420, 1807, ... (OEIS A039506),
and the corresponding numbers of odd primes that are
not cluster 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, p. 44) 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, p. 45) also show that
the sum of the reciprocals
of the cluster primes is finite.
