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

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

Author

César Eliud Lozada, Sep 01 2012

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Wikipedia, Harmonic Mean

Crossrefs

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

Programs

Maple

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;

Mathematica

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 *)