A216124 -id:A216124 - cp's OEIS frontend

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

César Eliud Lozada, Sep 01 2012

nonn

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

			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.

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

Cf. A006562, A216124.

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

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 *)

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 */

A216101 Primes which are the integer harmonic mean of the previous prime and the following prime.

Original entry on oeis.org

13, 19, 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, 1429, 1433

Offset: 1

César Eliud Lozada, Sep 01 2012

nonn

The harmonic mean of N numbers p1,p2,..,pN is the number H whose reciprocal is the arithmetic mean of the reciprocals of p1,p2,..,pN; that is to say, 1/H = ((1/p1)+(1/p2)+..+(1/pN))/N.
So, for two quantities p1 and p2, their harmonic mean may be written as H=(2*p1*p2)/(p1+p2).
The harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

			The primes before and after the prime p=13 are p1=11 and p2=17. So, the harmonic mean of p1 and p2 is 2*11*17/(11+17)=13.35714285... whose integer part is p=13. Then p=13 belongs to the sequence.
The primes before and after the prime p=17 are p1=13 and p2=19. The harmonic mean of p1 and p2 is 2*13*19/(13+19)=15.4375, having 15 as its integer part. Therefore, as 15<>p, p=17 is not in the sequence.

Wikipedia, Harmonic Mean

Cf. A006562, A001008, A102928, A216098, A216124.

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

t = {}; Do[p = Prime[n]; If[Floor[HarmonicMean[{Prime[n - 1], Prime[n + 1]}]] == p, AppendTo[t, p]], {n, 2, 200}]; t (* T. D. Noe, Sep 04 2012 *)

A349793 Primes which are the nearest integer to the harmonic mean of the previous prime and the following prime.

Original entry on oeis.org

3, 7, 13, 23, 47, 89, 157, 173, 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

Hugo Pfoertner, Nov 30 2021

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A006562, A046869, A051635, A075540, A216124.

Select[Partition[Prime[Range[750]],3,1],Round[HarmonicMean[{#[[1]],#[[3]]}]]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Dec 29 2024 *)

a349793(limit) = {my(p1=2,p2=3); forprime(p3=5, limit, my(hm=round((2*p1*p3)/(p1+p3))); if(p2==hm, print1(p2,", ")); p1=p2;p2=p3)};
a349793(5500)

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A216098 Primes that are equal to the floor of the geometric mean of the previous prime and the following prime.

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A216101 Primes which are the integer harmonic mean of the previous prime and the following prime.

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A349793 Primes which are the nearest integer to the harmonic mean of the previous prime and the following prime.

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