A051501 Bertrand primes III: a(n+1) is the smallest prime > 2^a(n).

2, 5, 37, 137438953481

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

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

Examples

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

References

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

Links

• 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.

Crossrefs

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

Extensions

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