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%I #31 Feb 16 2025 08:33:20
%S 5043,2417158053779,5245728941618725066052704993134,
%T 215872416866954281715178071724040762825421437510476267629647193878371
%N Round(2^(m-n-2)/(m*log(8))), where m = 2^n - n - 2.
%C a(9) has 145 digits and is too large to include.
%C Conjecture: a(n) < f(n) = number of primes of the form k*2^(n+2) + 1 with k odd that exist between a = 2^(n+2) + 1 and b = floor((2^(2^n) + 1)/(3*2^(n+2) + 1)).
%C For comparison, f(5) = 5746.
%C If the extended Riemann hypothesis is true, then for every fixed epsilon > 0, f(n) = Li(b)/(a - 1) + O(b^(1/2 + epsilon)), where Li(b) = integral(2..b, dt/log(t)).
%D P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 57-58.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>
%Y Cf. A000215, A016631, A023394, A046052.
%K nonn,easy,changed
%O 5,1
%A _Arkadiusz Wesolowski_, Oct 01 2013