A068828 Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).

2, 3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643

Offset: 1

Author

Amarnath Murthy, Mar 08 2002

Keywords

• easy
• nonn

Comments

Or, bad primes (version 1): primes not in A046869. - Jonathan Vos Post, Aug 20 2007

The idea can be extended by defining a geometrically weak prime of order k to be a prime which is less than the geometric mean of r neighbors on both sides for all r = 1 to k and not true for r = k+1. A similar extension could be defined for the sequence A051635.

It is easy to show that, except for the twin prime pair (3,5), the larger prime of every twin prime pair is in this sequence. The smaller prime of the pair is always in A046869. - T. D. Noe, Feb 19 2008

Examples

			23 belongs to this sequence as 23^2 = 529 < 19*29 = 551.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A051634, A051635, A006562, A000040, A046869.

Programs

Mathematica

Join[{2}, Prime[Select[Range[2, 120], Prime[ # ]^2 <= Prime[ # - 1]*Prime[ # + 1]&]]] (* Stefan Steinerberger, Aug 21 2007 *)
Join[{2},Transpose[Select[Partition[Prime[Range[500]],3,1],#[[2]]< GeometricMean[ {#[[1]],#[[3]]}]&]][[2]]] (* Harvey P. Dale, Apr 05 2014 *)

Formula

prime(k)^2 <= prime(k-1)*prime(k+1).

Extensions

Corrected and extended by Stefan Steinerberger, Aug 21 2007

Edited by N. J. A. Sloane, Feb 19 2008