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A216098 Primes that are equal to the floor of the geometric mean of the previous prime and the following prime. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A113911 A051635 A136009 * A310262 A310263 A056531

Adjacent sequences:  A216095 A216096 A216097 * A216099 A216100 A216101

KEYWORD

nonn

AUTHOR

César Eliud Lozada, Sep 01 2012

STATUS

approved

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Last modified February 28 06:55 EST 2020. Contains 332321 sequences. (Running on oeis4.)