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A051501 Bertrand primes III: a(n+1) is the smallest prime > 2^a(n). 2
2, 5, 37, 137438953481 (list; graph; refs; listen; history; text; internal format)



The terms in the sequence are floor(2^b), floor(2^2^b), floor(2^2^2^b), ..., where b is approximately 1.2516475977905.

The existence of b is a consequence of Bertrand's postulate.

a(5) is much larger than the largest known prime, which is currently only 2^32582657-1. - T. D. Noe, Oct 18 2007

This sequence is of course not computed from b; rather b is more precisely computed by determining the next term in the sequence.

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


R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Exercise 4.19.


Table of n, a(n) for n=1..4.

Pierre Dusart, Explicit estimates of some functions over primes, Ramanujan J. Vol 45 (2016), pp. 227-251.

E. M. Wright, A prime-representing function, Amer. Math. Monthly, 58 (1951), 616-618.


The smallest prime after 2^5 = 32 is 37, so a(5) = 37.


Cf. A006992 (Bertrand primes), A079614 (Bertrand's constant), A227770 (Bertrand primes II).

Sequence in context: A210482 A053609 A036780 * A221681 A347070 A290711

Adjacent sequences: A051498 A051499 A051500 * A051502 A051503 A051504




Jud McCranie


Although the exact value of the next term is not known, it has 41373247571 digits.

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]

Edited by Franklin T. Adams-Watters, Aug 10 2009

Reference and bounds on next term from Charles R Greathouse IV, Oct 27 2010

Name clarified by Jonathan Sondow, Aug 02 2013



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Last modified December 9 04:13 EST 2022. Contains 358698 sequences. (Running on oeis4.)