Balanced Primes

Balanced Primes  are primes which are the mean of the next primes number and the previous prime number. This can be expressed as , where   is the nth prime number and   and   are the previous prime number and the next prime number, respectively. 5 is the first balanced prime, as 3+7/2 =5. It is thought, although not proven, that there are infinitely many balanced primes. In theory, in order to be a balanced prime, the difference between the nth prime number and the n+1th prime number can be any even number, as long as it is the same as the gap between the nth and the n-1th. Most of said even numbers are multiples of three, as a result of the pigeonhole principle. If the difference is not a multiple of three, one of the three numbers n-x, n or n+x will be a multiple of three.

Examples

 * 53 is a balanced prime. The previous prime number is 47, while the next prime number is 59, both of which are 6 numbers away from 53.
 * 5 is a balanced prime. The pervious prime number is 3, while the next prime number is 7, both of which are 2 numbers away from 5.

The First Few Balanced Primes
5

53

157

173

211

257

263

373

563

593

607

653