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

Author

Vladimir Joseph Stephan Orlovsky, May 27 2010

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A069175, A014574, A069142, A099349.

Programs

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 *)

Formula

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