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

1,1

Comments

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

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A216101, A090076.

Sequence in context: A019363 A288890 A120384 * A096505 A214680 A141802

Adjacent sequences: A216121 A216122 A216123 * A216125 A216126 A216127

Keyword

nonn

Author

César Eliud Lozada, Sep 01 2012

Extensions

More terms from Michel Marcus, Apr 08 2015

Status

approved