An interprime is the average of consecutive (but not necessarily twin) odd primes. The first few terms are 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, ... (OEIS A024675).
The first few even interprimes are 4, 6, 12, 18, 26, 30, 34, 42, 50, 56, 60, ...
(OEIS A072568), and the first few odd ones
are 9, 15, 21, 39, 45, 69, 81, 93, 99, ... (OEIS A072569).

Interprimes cannot themselves be prime (since otherwise there would exist a prime between consecutive primes, which is impossible by definition).

The sum

has zeros at almost integer approximations of the interprimes, with the single additional point 5/2 (D. Tisdale, pers. comm.,
Sep. 8, 2008).