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%I #16 Apr 22 2021 17:56:35
%S 23,37,67,233,277,631,1039,1283,1297,1307,1613,1693,1709,2099,2137,
%T 2333,2719,2797,3271,3533,3547,3571,3923,4027,4253,4523,4643,4793,
%U 5483,5507,5647,6563,7321,8831,8849,9007,9029,10061,10079,10289,10513,12049,13687
%N Lonely non-twin primes: non-twins sandwiched between two pairs of twins.
%H Zak Seidov, <a href="/A068016/b068016.txt">Table of n, a(n) for n = 1..1000</a>
%H Brian Hopkins, <a href="http://ecajournal.haifa.ac.il/Volume2021/ECA2021_S1H1.pdf">Euler's Enumerations</a>, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.
%e 37 is in the sequence because it is adjacent to two pairs of twins (29,31 and 41,43). 47 is not because the primes adjacent to it are 43 and 53 and although 43 is a twin, 53 is not.
%t PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k]; lst={};Do[p=Prime[n];If[ !PrimeQ[p-2]&&PrimeQ[PrimePrev[p]-2]&&!PrimeQ[p+2]&&PrimeQ[PrimeNext[p]+2],AppendTo[lst,p]],{n,7!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2009 *)
%t m = 2000; #[[3]] & /@ Select[Partition[Prime[Range[7, m]], 5, 1], #[[2]] - #[[1]] == #[[5]] - #[[4]] == 2 &] (* _Zak Seidov_, Nov 25 2012 *)
%Y Cf. A001097, A069453.
%K nonn
%O 1,1
%A _Neil Fernandez_, Mar 22 2002
%E More terms from _Vladimir Joseph Stephan Orlovsky_, Dec 04 2009
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