balanced prime

If for the given $n$th prime $p_n$ the equality

$$p_n=\frac{1}{3}\sum _{i=n-1}^{n+1}{p}_i$$

is true, then $p_n$ is said to be a balanced prime. That is, the arithmetic mean of the given prime, the prime immediately below and the one immediately above, is equal to the middle prime. The first few are 5, 53, 157, 173, 211, 257, 263, 373, listed in A006562 of Sloane’s OEIS. As of 2006, the largest known balanced prime is $197418203\times 2^{25000}-1$, discovered by David Broadhurst and François Morain using FastECPP and PrimeForm.