A058249 (Smallest prime >= 2^n) - (largest prime <= 2^n).

0, 2, 4, 4, 6, 6, 4, 6, 12, 10, 14, 6, 18, 30, 22, 16, 30, 8, 22, 10, 26, 18, 24, 46, 74, 20, 68, 60, 14, 38, 12, 20, 26, 66, 84, 36, 34, 52, 30, 102, 48, 26, 86, 24, 114, 36, 120, 80, 150, 82, 150, 68, 116, 192, 58, 86, 22, 96, 186, 126, 16, 192, 54, 72, 180, 14, 22, 56

Offset: 1

Author

Labos Elemer, Dec 05 2000

Keywords

• nonn

Comments

This sequence gives the gap between consecutive primes on either side of 2^n. The average gap between primes near 2^n should be about g=n*log(2). Cramer's conjecture would allow gaps to be as large as about g^2. - T. D. Noe, Jul 17 2007

Examples

			n = 1: a(1) = 2 - 2 = 0,
n = 9: a(9) = 521 - 509 = 12.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10087 (first 5000 terms from T. D. Noe).

Crossrefs

Cf. A013603, A013597, A014210, A014234, A038804.

Programs

Maple

a := n -> if n > 1 then nextprime(2^n)-prevprime(2^n) else 0 fi; [seq( a(i), i=1..256)]; # Maple's next/prevprime functions use strict inequalities and therefore would not yield the correct difference for n=1. Alternatively, a(n) = nextprime(2^n-1)-prevprime(2^n+1);

Mathematica

Prepend[NextPrime[#]-NextPrime[#,-1]&/@(2^Range[2,70]),0] (* Harvey P. Dale, Jan 25 2011 *)
Join[{0}, Table[NextPrime[2^n] - NextPrime[2^n, -1], {n, 2, 70}]]

PARI

a(n)=nextprime(2^n)-precprime(2^n) \\ Charles R Greathouse IV, Sep 20 2016

Formula

a(n) = A014210(n) - A014234(n) = A013603(n) + A013597(n).

Extensions

Edited by M. F. Hasler, Feb 14 2017