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A068828
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Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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prime(k)^2 <= prime(k-1)*prime(k+1).
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EXAMPLE
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23 belongs to this sequence as 23^2 = 529 < 19*29 = 551.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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