A178421 Lower primes p1 in a twin pair such that sum of p1 and p2 yields average a1 of twin prime pairs and product of 2*a1 is another average of twin prime pairs.

211049, 248639, 253679, 410339, 507359, 605639, 1121189, 1138829, 1262099, 2162579, 2172869, 2277659, 4070219, 6305459, 7671509, 11659409, 12577109, 14203769, 14862119, 17472839, 18728639, 18798359, 20520569, 21140699

Offset

1,1

Comments

The definition means that a1/2, a1 and 2*a1 are all in A014574 (twin prime averages). - R. J. Mathar, Nov 02 2023

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A069175(n)-1. - R. J. Mathar, Nov 02 2023

Example

211049 is a term since 211049 and 211051 are twin primes; 211049 + 211051 = 422100 is an average of twin primes, and 2*422100 = 844200 is another average of twin primes.

Mathematica

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; a1=p1+p2; a2=2*a1; If[p2-p1==2&&PrimeQ[a1-1]&&PrimeQ[a1+1]&&PrimeQ[a2-1]&&PrimeQ[a2+1], AppendTo[lst, p1]], {n, 10!}]; lst
atpQ[{a_, b_}]:=Module[{m=a+b}, b-a==2&&AllTrue[m+{1, -1}, PrimeQ] && AllTrue[ 2m+{1, -1}, PrimeQ]]; Select[Partition[Prime[Range[134*10^4]], 2, 1], atpQ][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2019 *)

Crossrefs

Cf. A069175, A014574, A069142, A099349.

Sequence in context: A216593 A172825 A072760 * A069175 A340158 A097021

Adjacent sequences: A178418 A178419 A178420 * A178422 A178423 A178424

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 27 2010

Status

approved