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