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%I #20 Jun 19 2022 18:30:27
%S 2,3,7,13,19,23,31,43,47,61,73,83,89,103,109,113,131,139,151,167,181,
%T 193,199,229,233,241,271,283,293,313,317,337,349,353,359,383,389,401,
%U 409,421,433,443,449,463,467,491,503,509,523,547,571,577,601,619,643
%N Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).
%C Or, bad primes (version 1): primes not in A046869. - _Jonathan Vos Post_, Aug 20 2007
%C 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.
%C 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
%H T. D. Noe, <a href="/A068828/b068828.txt">Table of n, a(n) for n=1..10000</a>
%F prime(k)^2 <= prime(k-1)*prime(k+1).
%e 23 belongs to this sequence as 23^2 = 529 < 19*29 = 551.
%t Join[{2}, Prime[Select[Range[2, 120], Prime[ # ]^2 <= Prime[ # - 1]*Prime[ # + 1]&]]] (* _Stefan Steinerberger_, Aug 21 2007 *)
%t Join[{2},Transpose[Select[Partition[Prime[Range[500]],3,1],#[[2]]< GeometricMean[ {#[[1]],#[[3]]}]&]][[2]]] (* _Harvey P. Dale_, Apr 05 2014 *)
%Y Cf. A051634, A051635, A006562, A000040, A046869.
%K easy,nonn
%O 1,1
%A _Amarnath Murthy_, Mar 08 2002
%E Corrected and extended by _Stefan Steinerberger_, Aug 21 2007
%E Edited by _N. J. A. Sloane_, Feb 19 2008
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