A046869 Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 227, 239, 251, 257, 263, 269, 277, 281, 307, 311, 331, 347, 367, 373, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also called geometrically strong primes. - Amarnath Murthy, Mar 08 2002

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

It appears that a(n) ~ 2*prime(n). - Thomas Ordowski, Jul 25 2012

Conjecture: primes p(n) such that 2*p(n) >= p(n-1) + p(n+1). - Thomas Ordowski, Jul 25 2012

Probably {3,7,23} U {good primes} = {primes p(n) > 2/(1/p(n-1) + 1/p(n+1))}. - Thomas Ordowski, Jul 27 2012

Except for A001359(1), A001359 is a subsequence. - Chai Wah Wu, Sep 10 2019

Examples

			37 is a member as 37^2 = 1369 > 31*41 = 1271.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Section A14.

Links

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

Crossrefs

Cf. A001359, A006562, A028388, A046868, A051634, A051635, A068828.

Programs

Magma

[NthPrime(n): n in [2..100] | NthPrime(n)^2 gt NthPrime(n-1)*NthPrime(n+1)]; // Bruno Berselli, Oct 23 2012

Maple

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;

Mathematica

Do[ If[ Prime[n]^2 > Prime[n - 1]*Prime[n + 1], Print[ Prime[n] ] ], {n, 2, 100} ]
Transpose[Select[Partition[Prime[Range[300]],3,1],#[[2]]^2>#[[1]]#[[3]]&]][[2]] (* Harvey P. Dale, May 13 2012 *)
Select[Prime[Range[2, 100]], #^2 > NextPrime[#]*NextPrime[#, -1] &] (* Jayanta Basu, Jun 29 2013 *)

PARI

forprime(n=o=p=3,999,o+0<(o=p)^2/(p=n) & print1(o", "))
isA046869(p)={ isprime(p) & p^2>precprime(p-1)*nextprime(p+1) } \\ M. F. Hasler, Jun 15 2011

Extensions

Corrected and extended by Robert G. Wilson v, Dec 06 2000

Edited by N. J. A. Sloane at the suggestion of Giovanni Resta, Aug 20 2007