Balanced prime

In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, the ${\displaystyle n}$th prime number ${\displaystyle p_{n}}$ is a balanced prime if

${\displaystyle p_{n}={{p_{n-1}+p_{n+1}} \over 2}.}$

For example, 53 is the sixteenth prime; the fifteenth and seventeenth primes, 47 and 59, add up to 106, and half of that is 53; thus 53 is a balanced prime.