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%I #28 Oct 16 2023 12:03:34
%S 2,5,37,137438953481
%N Bertrand primes III: a(n+1) is the smallest prime > 2^a(n).
%C The terms in the sequence are floor(2^b), floor(2^2^b), floor(2^2^2^b), ..., where b is approximately 1.2516475977905.
%C The existence of b is a consequence of Bertrand's postulate.
%C a(5) is much larger than the largest known prime, which is currently only 2^32582657-1. - _T. D. Noe_, Oct 18 2007
%C This sequence is of course not computed from b; rather b is more precisely computed by determining the next term in the sequence.
%C Robert Ballie comments that the next term is known to be 2.80248435135615213561103452115581... * 10^41373247570 via Dusart 2016, improving on my 2010 result in the Extensions section. - _Charles R Greathouse IV_, Aug 11 2020
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Exercise 4.19.
%H Pierre Dusart, <a href="https://dx.doi.org/10.1007/s11139-016-9839-4">Explicit estimates of some functions over primes</a>, Ramanujan J. Vol 45 (2016), pp. 227-251.
%H E. M. Wright, <a href="http://www.jstor.org/stable/2306356">A prime-representing function</a>, Amer. Math. Monthly, 58 (1951), 616-618.
%e The smallest prime after 2^5 = 32 is 37, so a(5) = 37.
%Y Cf. A006992 (Bertrand primes), A079614 (Bertrand's constant), A227770 (Bertrand primes II).
%K nonn
%O 1,1
%A _Jud McCranie_
%E Although the exact value of the next term is not known, it has 41373247571 digits.
%E Next term is 2.8024843513561521356110...e41373247570, where the next digit is 3 or 4. Under the Riemann hypothesis, the first 20686623775 digits are known. [From _Charles R Greathouse IV_, Oct 27 2010]
%E Edited by _Franklin T. Adams-Watters_, Aug 10 2009
%E Reference and bounds on next term from _Charles R Greathouse IV_, Oct 27 2010
%E Name clarified by _Jonathan Sondow_, Aug 02 2013
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