A214497 - OEIS

%I #14 Dec 09 2012 09:18:39

%S 0,6,3,9,9,6,3,93,3,54,18,96,213,297,1206,258,312,201,261,1206,1158,

%T 396,1062,216,708,762,816,678,3579,762,831,2106,4734,576,333,633,213,

%U 2766,363,2454,1464,2007,4551,3183,1497,4899,198,66,9984,2847,276,3051

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

%C Conjecture : there is always one such k for each n>0.

%C As N increases, the average of a(n)/n^2 over n=1 to N appears to approach 1.1

%H Pierre CAMI, <a href="/A214497/b214497.txt">Table of n, a(n) for n = 1..500</a>

%p A214497 := proc(n)

%p     local k;

%p     for k from 0 do

%p         p := (3^n-k)*2^n-1 ;

%p         if isprime(p) and isprime(p+2) then

%p             return k;

%p         end if;

%p     end do:

%p end proc:

%p seq(A214497(n),n=1..80) ; # _R. J. Mathar_, Jul 23 2012

%t 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 (PFGW64 and SCRIPTIFY)

%o SCRIPT

%o DIM nn,0

%o DIM kk

%o DIM jj

%o DIMS tt

%o OPENFILEOUT myfile,a(n).txt

%o OPENFILEOUT myf,b(n).txt

%o LABEL loopn

%o SET nn,nn+1

%o SET jj,0

%o IF nn>500 THEN END

%o SET kk,-1

%o LABEL loopk

%o SET kk,kk+1

%o SETS tt,%d,%d\,;nn;kk

%o PRP (3^nn-kk)*2^nn-1,tt

%o IF ISPRP THEN GOTO a

%o IF ISPRIME THEN GOTO a

%o GOTO loopk

%o LABEL a

%o SET jj,jj+1

%o PRP (3^nn-kk)*2^nn+1,tt

%o IF ISPRP THEN GOTO d

%o IF ISPRIME THEN GOTO d

%o GOTO loopk

%o LABEL d

%o WRITE myfile,tt

%o SETS tt,%d,%d\,;nn;jj

%o WRITE myf,tt

%o GOTO loopn

%Y Cf. A214495-A214498.

%K nonn

%O 1,2

%A _Pierre CAMI_, Jul 20 2012