A216098 Primes that are equal to the floor of the geometric mean of the previous prime and the following prime.

3, 7, 13, 19, 23, 43, 47, 83, 89, 103, 109, 131, 167, 193, 229, 233, 313, 349, 353, 359, 383, 389, 409, 443, 449, 463, 503, 643, 647, 677, 683, 691, 709, 797, 823, 859, 883, 919, 941, 971, 983, 1013, 1093, 1097, 1109, 1171, 1193, 1217, 1279, 1283, 1303, 1373

Offset: 1

Author

César Eliud Lozada, Sep 01 2012

Keywords

• nonn

Comments

The geometric mean of two primes p and q is sqrt(pq).

Except for 3, these are also primes prime(k) such that (prime(k-1) + prime(k+1))/2 = prime(k)+1, verified up to k=50000. - Richard R. Forberg, Jun 29 2015

Primes prime(k) such that (prime(k)+1)^2 > prime(k-1)*prime(k+1) > prime(k)^2. - Robert Israel, Jul 10 2015

Examples

			The primes before and after the prime 3 are 2 and 5, so the geometric mean is sqrt(2*5)=sqrt(10)=3.16227766..., whose integer part is 3. Therefore 3 is in the sequence.
The primes before and after the prime 11 are 7 and 13. The geometric mean of 7 and 13 is sqrt(7*13)=9.539392... whose integer part is 9 and not 11, hence 11 is not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006562, A216124.

Programs

Maple

A := {}: for n from 2 to 1000 do p1 := ithprime(n-1); p := ithprime(n); p2 := ithprime(n+1); if p = floor(sqrt(p1*p2)) then A := `union`(A, {p}) end if end do; A := A

Mathematica

t = {}; Do[p = Prime[n]; If[Floor[GeometricMean[{Prime[n-1], Prime[n+1]}]] == p, AppendTo[t, p]], {n, 2, 200}]; t (* T. D. Noe, Sep 04 2012 *)

PARI

first(m)=my(v=vector(m)); t=2; k=1; while(k<=m, p=prime(t);if(p==floor(sqrt(prime(t-1)*prime(t+1))), v[k]=p;k++); t++);v; /* Anders Hellström, Aug 03 2015 */