A216124 Primes which are the nearest integer to the geometric mean of the previous prime and the following prime.

3, 5, 7, 23, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393

Offset: 1

Author

César Eliud Lozada, Sep 01 2012

Keywords

• nonn

Comments

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

Examples

			The prime before 3 is 2 and the prime after 3 is 5. 2 * 5 = 10 and the geometric mean of 2 and 5 is therefore sqrt(10) = 3.16227766..., which rounds to 3. Therefore 3 is in the sequence.
The geometric mean of 7 and 13 is 9.539392... which rounds up to 10, well short of 11, hence 11 is not in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A216101, A090076.

Programs

Maple

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

Mathematica

Prime[Select[Range[2, 700], Prime[#] == Round[Sqrt[Prime[# - 1] Prime[# + 1]]] &]] (* Alonso del Arte, Sep 01 2012 *)
Select[Partition[Prime[Range[750]],3,1],Round[GeometricMean[{#[[1]],#[[3]]}]]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Feb 28 2024 *)

PARI

lista(nn) = forprime (p=2, nn, if (round(sqrt(precprime(p-1)*nextprime(p+1))) == p, print1(p, ", "))); \\ Michel Marcus, Apr 08 2015

Python

from math import isqrt
from itertools import islice
from sympy import nextprime, prevprime
def A216124_gen(startvalue=3): # generator of terms >= startvalue
    q = max(3,nextprime(startvalue-1))
    p = prevprime(q)
    r = nextprime(q)
    while True:
        if q == (m:=isqrt(k:=p*r))+(k-m*(m+1)>=1):
            yield q
        p, q, r = q, r, nextprime(r)
A216124_list = list(islice(A216124_gen(),20)) # Chai Wah Wu, Jun 19 2024

Extensions

More terms from Michel Marcus, Apr 08 2015