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A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.

%I A102723 #22 Oct 16 2021 05:11:52
%S A102723 5,23,23,53,53,211,211,211,211,211,211,1847,1847,2179,2179,2179,2179,
%T A102723 3967,3967,16033,16033,16033,16033,24281,24281,24281,24281,24281,
%U A102723 24281,38501,38501,38501,38501,38501,38501,38501,38501,38501,38501,58831
%N A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.
%C A102723 a(2n+1)=a(2n). - _Robert G. Wilson v_, Feb 22 2005
%C A102723 Using Dirichlet's theorem, Sierpiński (1948) proved that a(n) exists for all n > 0. He noted that a(n) is a non-twin prime (A007510), except for a(1) = 5. - _Jonathan Sondow_, Oct 27 2017
%H A102723 David A. Corneth, <a href="/A102723/b102723.txt">Table of n, a(n) for n = 1..479</a> (first 97 terms from Harvey P. Dale)
%H A102723 W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm1/cm131.pdf"> Remarque sur la répartition des nombres premiers</a>, Colloq. Math., 1 (1948), 193-194.
%t A102723 f[n_] := Block[{k = 1}, While[ Union[ PrimeQ /@ Sort[ Flatten[ Table[{Prime[k] - i, Prime[k] + i}, {i, n}]]]] != {False}, k++ ]; Prime[k]]; Table[ f[n], {n, 40}] (* _Robert G. Wilson v_, Feb 22 2005 *)
%t A102723 cmpgap[n_]:=Module[{p=Prime[n]},Min[p-NextPrime[p,-1],NextPrime[p]-p]]; Module[{nn=10000,prs},prs=Table[{Prime[n],cmpgap[n]},{n,nn}];Table[ SelectFirst[ prs,#[[2]]>=k&],{k,2,50}]][[All,1]] (* _Harvey P. Dale_, Oct 15 2021 *)
%Y A102723 Cf. A007510, A023186.
%K A102723 nonn
%O A102723 1,1
%A A102723 _Ray G. Opao_, Feb 06 2005
%E A102723 a(12)-a(40) from _Robert G. Wilson v_, Feb 22 2005