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A214495 Smallest k>0 such that (3^n-k)*3^n-1 and (3^n-k)*3^n+1 are a twin prime pair.

%I A214495 #14 Sep 05 2012 12:11:15
%S A214495 1,1,11,19,7,1,23,31,53,49,47,139,49,101,97,399,87,281,37,329,893,497,
%T A214495 203,883,213,1171,633,593,1747,349,3843,479,2347,329,1921,1299,2933,
%U A214495 1467,3097,1943,1509,2077,2111,723,2913,2307,963,361,297,1249,1031,2153
%N A214495 Smallest k>0 such that (3^n-k)*3^n-1 and (3^n-k)*3^n+1 are a twin prime pair.
%C A214495 Conjecture : there is always one such k(n) for each n>0.
%C A214495 Heuristically, as N increases, the average of a(n)/n^2 over n=1 to N tends to 1.2
%H A214495 Pierre CAMI, <a href="/A214495/b214495.txt">Table of n, a(n) for n = 1..500</a>
%p A214495 A214495 := proc(n)
%p A214495     local k;
%p A214495     for k from 1 do
%p A214495         p := (3^n-k)*3^n-1 ;
%p A214495         if isprime(p) and isprime(p+2) then
%p A214495             return k;
%p A214495         end if;
%p A214495     end do:
%p A214495 end proc: # _R. J. Mathar_, Jul 23 2012
%t A214495 sk[n_]:=Module[{k=1,n3=3^n},While[!PrimeQ[(n3-k)*n3-1]||!PrimeQ[(n3-k)* n3+1], k++];k]; Array[sk,60] (* _Harvey P. Dale_, Sep 05 2012 *)
%o A214495 (PFGW64 and SCRIPTIFY)
%o A214495 SCRIPT
%o A214495 DIM nn,0
%o A214495 DIM kk
%o A214495 DIM jj
%o A214495 DIMS tt
%o A214495 OPENFILEOUT myfile,a(n).txt
%o A214495 OPENFILEOUT myf,b(n).txt
%o A214495 LABEL loopn
%o A214495 SET nn,nn+1
%o A214495 SET jj,0
%o A214495 IF nn>500 THEN END
%o A214495 SET kk,-1
%o A214495 LABEL loopk
%o A214495 SET kk,kk+1
%o A214495 SETS tt,%d,%d\,;nn;kk
%o A214495 PRP (3^nn-kk)*3^nn-1,tt
%o A214495 IF ISPRP THEN GOTO a
%o A214495 IF ISPRIME THEN GOTO a
%o A214495 GOTO loopk
%o A214495 LABEL a
%o A214495 SET jj,jj+1
%o A214495 PRP (3^nn-kk)*3^nn+1,tt
%o A214495 IF ISPRP THEN GOTO d
%o A214495 IF ISPRIME THEN GOTO d
%o A214495 GOTO loopk
%o A214495 LABEL d
%o A214495 WRITE myfile,tt
%o A214495 SETS tt,%d,%d\,;nn;jj
%o A214495 WRITE myf,tt
%o A214495 GOTO loopn
%Y A214495 Cf. A214496.
%K A214495 nonn
%O A214495 1,3
%A A214495 _Pierre CAMI_, Jul 19 2012