Balanced prime

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A balanced prime is a prime number that is equal to the arithmetic mean of the nearest prime above and below. Or to put it algebraically, given a prime number $p n$ , where n is its index in the ordered set of prime numbers, $p_n = {{p_{n - 1} + p_{n + 1}} \over 2}$ . The first few balanced primes are

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 (sequence A006562 in OEIS)

For example, 53 is the sixteenth prime. The fifteenth and seventeenth primes, 47 and 59, add up to 106, which is twice 53, thus 53 is a balanced prime.

When 1 was considered a prime number, 2 would have correspondingly been considered the first balanced prime since $2 = {(1 + 3) \over 2}$ .

It is conjectured that there are infinitely many balanced primes.
3 consecutive primes in arithmetic progression is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. As of 2005 the largest known CPAP-3 has 7535 digits found by David Broadhurst and François Morain[1]:
$p_n = 197418203 \times 2^{25000} - 1, p_{n-1} = p_n-6090, p_{n+1} = p_n+6090$
The value of n is not known.