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A214497 Smallest k>=0 such that (3^n-k)*2^n-1 and (3^n-k)*2^n+1 are a twin prime pair.

%I A214497 #14 Dec 09 2012 09:18:39
%S A214497 0,6,3,9,9,6,3,93,3,54,18,96,213,297,1206,258,312,201,261,1206,1158,
%T A214497 396,1062,216,708,762,816,678,3579,762,831,2106,4734,576,333,633,213,
%U A214497 2766,363,2454,1464,2007,4551,3183,1497,4899,198,66,9984,2847,276,3051
%N A214497 Smallest k>=0 such that (3^n-k)*2^n-1 and (3^n-k)*2^n+1 are a twin prime pair.
%C A214497 Conjecture : there is always one such k for each n>0.
%C A214497 As N increases, the average of a(n)/n^2 over n=1 to N appears to approach 1.1
%H A214497 Pierre CAMI, <a href="/A214497/b214497.txt">Table of n, a(n) for n = 1..500</a>
%p A214497 A214497 := proc(n)
%p A214497     local k;
%p A214497     for k from 0 do
%p A214497         p := (3^n-k)*2^n-1 ;
%p A214497         if isprime(p) and isprime(p+2) then
%p A214497             return k;
%p A214497         end if;
%p A214497     end do:
%p A214497 end proc:
%p A214497 seq(A214497(n),n=1..80) ; # _R. J. Mathar_, Jul 23 2012
%t A214497 sk[n_]:=Module[{k=0,c},c=(3^n-k)2^n;While[!PrimeQ[c-1] || !PrimeQ[c+1],k++;c=(3^n-k)2^n];k]; Array[sk,60] (* _Harvey P. Dale_, Dec 09 2012 *)
%o A214497 (PFGW64 and SCRIPTIFY)
%o A214497 SCRIPT
%o A214497 DIM nn,0
%o A214497 DIM kk
%o A214497 DIM jj
%o A214497 DIMS tt
%o A214497 OPENFILEOUT myfile,a(n).txt
%o A214497 OPENFILEOUT myf,b(n).txt
%o A214497 LABEL loopn
%o A214497 SET nn,nn+1
%o A214497 SET jj,0
%o A214497 IF nn>500 THEN END
%o A214497 SET kk,-1
%o A214497 LABEL loopk
%o A214497 SET kk,kk+1
%o A214497 SETS tt,%d,%d\,;nn;kk
%o A214497 PRP (3^nn-kk)*2^nn-1,tt
%o A214497 IF ISPRP THEN GOTO a
%o A214497 IF ISPRIME THEN GOTO a
%o A214497 GOTO loopk
%o A214497 LABEL a
%o A214497 SET jj,jj+1
%o A214497 PRP (3^nn-kk)*2^nn+1,tt
%o A214497 IF ISPRP THEN GOTO d
%o A214497 IF ISPRIME THEN GOTO d
%o A214497 GOTO loopk
%o A214497 LABEL d
%o A214497 WRITE myfile,tt
%o A214497 SETS tt,%d,%d\,;nn;jj
%o A214497 WRITE myf,tt
%o A214497 GOTO loopn
%Y A214497 Cf. A214495-A214498.
%K A214497 nonn
%O A214497 1,2
%A A214497 _Pierre CAMI_, Jul 20 2012