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%I #57 Sep 08 2022 08:44:56
%S 5,11,17,29,37,41,53,59,67,71,79,97,101,107,127,137,149,157,163,173,
%T 179,191,197,211,223,227,239,251,257,263,269,277,281,307,311,331,347,
%U 367,373,379,397,419,431,439,457,461,479,487,499,521,541
%N Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).
%C Also called geometrically strong primes. - _Amarnath Murthy_, Mar 08 2002
%C The idea can be extended by defining a geometrically strong prime of order k to be a prime that is greater than the geometric mean of r neighbors on both sides for all r = 1 to k but not for r = k+1. Similar generalizations can be applied to the sequence A051634. - _Amarnath Murthy_, Mar 08 2002
%C It appears that a(n) ~ 2*prime(n). - _Thomas Ordowski_, Jul 25 2012
%C Conjecture: primes p(n) such that 2*p(n) >= p(n-1) + p(n+1). - _Thomas Ordowski_, Jul 25 2012
%C Probably {3,7,23} U {good primes} = {primes p(n) > 2/(1/p(n-1) + 1/p(n+1))}. - _Thomas Ordowski_, Jul 27 2012
%C Except for A001359(1), A001359 is a subsequence. - _Chai Wah Wu_, Sep 10 2019
%D R. K. Guy, Unsolved Problems in Number Theory, Section A14.
%H T. D. Noe, <a href="/A046869/b046869.txt">Table of n, a(n) for n = 1..10000</a>
%e 37 is a member as 37^2 = 1369 > 31*41 = 1271.
%p with(numtheory): a := [ ]: P := [ ]: M := 300: for i from 2 to M do if p(i)^2>p(i-1)*p(i+1) then a := [ op(a),i ]; P := [ op(P),p(i) ]; fi; od: a; P;
%t Do[ If[ Prime[n]^2 > Prime[n - 1]*Prime[n + 1], Print[ Prime[n] ] ], {n, 2, 100} ]
%t Transpose[Select[Partition[Prime[Range[300]],3,1],#[[2]]^2>#[[1]]#[[3]]&]][[2]] (* _Harvey P. Dale_, May 13 2012 *)
%t Select[Prime[Range[2, 100]], #^2 > NextPrime[#]*NextPrime[#, -1] &] (* _Jayanta Basu_, Jun 29 2013 *)
%o (PARI) forprime(n=o=p=3,999,o+0<(o=p)^2/(p=n) & print1(o", "))
%o isA046869(p)={ isprime(p) & p^2>precprime(p-1)*nextprime(p+1) } \\ _M. F. Hasler_, Jun 15 2011
%o (Magma) [NthPrime(n): n in [2..100] | NthPrime(n)^2 gt NthPrime(n-1)*NthPrime(n+1)]; // _Bruno Berselli_, Oct 23 2012
%Y Cf. A001359, A006562, A028388, A046868, A051634, A051635, A068828.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E Corrected and extended by _Robert G. Wilson v_, Dec 06 2000
%E Edited by _N. J. A. Sloane_ at the suggestion of _Giovanni Resta_, Aug 20 2007
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