The first few balanced primes are 5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 (sequence A006562 in the OEIS).
It is conjectured that there are infinitely many balanced primes.
Three consecutive primes in arithmetic progression is sometimes called a CPAP-3.
As of 2023[update] the largest known CPAP-3 has 15004 decimal digits and was found by Serge Batalov.
The sequences of balanced primes of orders 2, 3, and 4 are A082077, A082078, and A082079 in the OEIS respectively.