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, given a prime number formula_1, where is its index in the ordered set of prime numbers, 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. Examples. The first few balanced primes are 5, 53, 157, 173 , 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103 . Infinitude. It is conjectured that there are infinitely many balanced primes. Three consecutive primes in arithmetic progression is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is: The value of "n" (its rank in the sequence of all primes) is not known. Generalization. The balanced primes may be generalized to the balanced primes of order "n". A balanced prime of order "n" is a prime number  that is equal to the arithmetic mean of the nearest "n" primes above and below. Algebraically, given a prime number formula_4, where "k" is its index in the ordered set of prime numbers, Thus, an ordinary balanced prime is a balanced prime of order 1. The sequences of balanced primes of orders 2, 3, and 4 are given as sequence in the OEIS, sequence in the OEIS, and sequence in the OEIS respectively.