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

1,1

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

Links

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

Example

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.

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 *)

Prog

(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 */

Crossrefs

Cf. A006562, A216124.

Sequence in context: A365241 A051635 A136009 * A310262 A310263 A056531

Adjacent sequences: A216095 A216096 A216097 * A216099 A216100 A216101

Keyword

nonn

Author

César Eliud Lozada, Sep 01 2012

Status

approved