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A205322 Smallest k>=0 such that (2^n-k)*2^n-1 and (2^n-k)*2^n+1 are a twin prime pair; or -1 if no such k exists.

%I A205322 #14 Mar 15 2013 12:33:20
%S A205322 0,1,-1,1,26,22,8,28,47,16,14,19,17,316,8,133,116,166,77,364,197,49,
%T A205322 647,1594,848,109,869,169,773,166,1274,466,512,694,644,733,401,1636,
%U A205322 662,184,71,3106,1157,346,332,2194,6179,7999,6023,6784,5612,1108,1001,649,197
%N A205322 Smallest k>=0 such that (2^n-k)*2^n-1 and (2^n-k)*2^n+1 are a twin prime pair; or -1 if no such k exists.
%C A205322 Conjecture: there is always at least one k>=0 unless n=3.
%H A205322 Pierre CAMI, <a href="/A205322/b205322.txt">Table of n, a(n) for n = 1..826</a>
%p A205322 A205322 := proc(n)
%p A205322     local a,p ;
%p A205322     for a from 0 to 2^n do
%p A205322          p := (2^n-a)*2^n-1 ;
%p A205322         if isprime(p) and isprime(p+2) then
%p A205322             return a;
%p A205322         end if;
%p A205322     end do:
%p A205322     return -1 ;
%p A205322 end proc: # _R. J. Mathar_, Jul 18 2012
%t A205322 Table[k = -1; While[k++; p = 4^n - k*2^n - 1; p > 0 && ! (PrimeQ[p] && PrimeQ[p + 2])]; If[p <= 0, -1, k], {n, 50}] (* _T. D. Noe_, Mar 15 2013 *)
%o A205322 (PFGW64 and SCRIPTIFY)
%o A205322 SCRIPT
%o A205322 DIM nn, 0
%o A205322 DIM kk
%o A205322 DIMS tt
%o A205322 OPENFILEOUT myfile, a(n).txt
%o A205322 LABEL loopn
%o A205322 SET nn, nn+1
%o A205322 IF nn==3 THEN SET nn,4
%o A205322 IF nn>826 THEN END
%o A205322 SET kk, -1
%o A205322 LABEL loopk
%o A205322 SET kk, kk+1
%o A205322 SETS tt, %d, %d\,; nn; kk
%o A205322 PRP (2^nn-kk)*2^nn-1, tt
%o A205322 IF ISPRP THEN GOTO a
%o A205322 IF ISPRIME THEN GOTO a
%o A205322 GOTO loopk
%o A205322 LABEL a
%o A205322 PRP (2^nn-kk)*2^nn+1, tt
%o A205322 IF ISPRP THEN GOTO d
%o A205322 IF ISPRIME THEN GOTO d
%o A205322 GOTO loopk
%o A205322 LABEL d
%o A205322 WRITE myfile, tt
%o A205322 GOTO loopn
%Y A205322 Cf. A191619, A191620, A205321.
%K A205322 sign
%O A205322 1,5
%A A205322 _Pierre CAMI_, Jul 14 2012