Zeros of a "bad sequence" with the pi function

[Tomasz Ordowski]

Hello Seq Fans!

Let a(n) be the largest 0 <= m < n such that 

2 pi(n) = pi(n-m) + pi(n+m).  Find the zeros. 

Conjecture: for n > 1, a(n) = 0 if and only if

n is a "good prime" of the type A127925.

These are "midpoint convex primes". 

See https://oeis.org/draft/A127925

Is this equivalence provable? 

Requires prior checking. 

Best, 

Tom Ordo