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%I #8 Aug 01 2020 10:31:22
%S 3,5,23,67,233,503,683,1013,1759,2099,2797,3169,10663,12391,12899,
%T 13487,15149,18583,20563,21881,25373,26237,26681,33613,36787,36943,
%U 41411,41443,43573,61547,63337,63841,68909,71999,75721,76367,76481,86677
%N Middle of 3 consecutive prime numbers, p1, p2, p3, such that p1*p2*p3*d1*d2 = average of twin prime pairs; d1 (delta) = p2 - p1, d2 (delta) = p3 - p2.
%H Amiram Eldar, <a href="/A153410/b153410.txt">Table of n, a(n) for n = 1..10000</a>
%e 2*3*5*1*2 = 60 and 60 +- 1 are primes.
%e 3*5*7*2*2 = 420 and 420 +- 1 are primes.
%e 19*23*29*4*6 = 304152 and 304152 +- 1 are primes.
%t lst={};Do[p1=Prime[n];p2=Prime[n+1];p3=Prime[n+2];d1=p2-p1;d2=p3-p2;a=p1*p2*p3*d1*d2;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,p2]],{n,8!}];lst
%t cpnQ[{a_,b_,c_}]:=Module[{x=Times@@Join[{a,b,c},Differences[ {a,b,c}]]}, AllTrue[ x+{1,-1},PrimeQ]]; Select[Partition[ Prime[Range[ 10000]],3,1], cpnQ][[All,2]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 01 2020 *)
%Y Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Dec 25 2008
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