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A160916 Averages of twin prime pairs which are a sum of averages of two consecutive twin prime pairs.

%I A160916 #11 Jul 12 2020 13:34:14
%S A160916 18,30,72,102,660,1092,1302,2082,2112,2382,2592,2970,4242,4272,4650,
%T A160916 4722,4932,5442,5520,6690,6762,7590,8220,9000,10332,10860,11160,11718,
%U A160916 11970,13002,16452,17658,18522,20232,22278,23040,23910,24180,24918
%N A160916 Averages of twin prime pairs which are a sum of averages of two consecutive twin prime pairs.
%C A160916 Values A014574(j) of the form A014574(k) + A014574(k+1).
%H A160916 Amiram Eldar, <a href="/A160916/b160916.txt">Table of n, a(n) for n = 1..10000</a>
%e A160916 18=A014574(4) is in the sequence because it equals A014574(2)+A014574(3) = 6+12.
%e A160916 30=A014574(5) is in the sequence because it equals A014574(3)+A014574(4) = 12+18.
%e A160916 72=A014574(8) is in the sequence because it equals A014574(5)+A014574(6) = 30+42.
%t A160916 PrimeNextTwinAverage[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k-1]||!PrimeQ[k+1], k++ ];k];lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],a=n;a+=PrimeNextTwinAverage[a]; If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,a]]],{n,8!}];lst
%t A160916 Select[Total/@Partition[Mean/@Select[Partition[Prime[Range[ 2000]],2,1], #[[2]] - #[[1]] ==2&],2,1],AllTrue[#+{1,-1},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 12 2020 *)
%Y A160916 Cf. A014574.
%K A160916 nonn
%O A160916 1,1
%A A160916 _Vladimir Joseph Stephan Orlovsky_, May 30 2009
%E A160916 Comment and examples edited by _R. J. Mathar_, Jul 31 2007