Balanced Prime

A balanced prime is a prime number that is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number, where n is its index in the ordered set of prime numbers,

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, half of which is 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

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. As of 2009 the largest known CPAP-3 with proven primes has 7535 digits found by David Broadhurst and François Morain:

The value of n is not known.

Other articles related to "prime, balanced prime":

9000 (number) - Selected Numbers in The Range 9000–9999
... cube number 9025 – 952, centered octagonal number 9029 – Sophie Germain prime 9045 – triangular number 9059 – Sophie Germain prime 9072 – decagonal number 9077 – Markov number 9091. 9870 – triangular number 9871 – balanced prime 9880 – tetrahedral number 9887 – safe prime 9899 – ISO 9899 standard for C programming language 9901 ...

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