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

1,1

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

References

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

Links

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A023489 A268307 A108294 * A028388 A277718 A067606

Adjacent sequences: A046866 A046867 A046868 * A046870 A046871 A046872

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved