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Least k such that k*2^n +/- 1 are twin primes.
20

%I #21 Sep 09 2023 06:48:40

%S 4,2,1,9,12,6,3,9,57,30,15,99,165,90,45,24,12,6,3,69,132,66,33,486,

%T 243,324,162,81,90,45,345,681,585,375,267,426,213,429,288,144,72,36,

%U 18,9,147,810,405,354,177,1854,927,1125,1197,666,333,519,1032,516,258,129,72

%N Least k such that k*2^n +/- 1 are twin primes.

%C Excluding the first three terms, all remaining terms have digital root 3, 6, or 9. - _J. W. Helkenberg_, Jul 24 2013

%D Richard Crandall and Carl Pomerance, 'Prime Numbers: A Computational Perspective,' Springer-Verlag, NY, 2001, page 12.

%H Pierre CAMI, <a href="/A063983/b063983.txt">Table of n, a(n) for n = 0..2300</a>

%e a(3) = 9 because 9*2^3=72 and 71 and 73 are twin primes.

%e n=6: a(6)=3, 64.3=192 and {191,193} are both primes; n=71: a(71)=630, 630*[2^71]=1487545442103938242314240 and {1487545442103938242314239, 1487545442103938242314241} are twin primes.

%t Table[Do[s=(2^j)*k; If[PrimeQ[s-1]&&PrimeQ[s+1],Print[{j,k]], {k,1,2*j^2],{j,0,100]; (*outprint of a[j]=k*)

%t Do[ k = 1; While[ ! PrimeQ[ k*2^n + 1 ] || ! PrimeQ[ k*2^n - 1 ], k++ ]; Print[ k ], {n, 0, 50} ]

%t f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2^n*k], k++ ];k];Table[f[n], {n, 60}] (* _Ray Chandler_, Jan 09 2009 *)

%Y Cf. A040040, A045753, A002822, A124065, A124518-A124522.

%Y Cf. A071256, A060210, A060256. For records see A125848, A125019.

%Y Cf. A076806 (requires odd k)

%K nonn

%O 0,1

%A _Robert G. Wilson v_, Sep 06 2001

%E More terms from _Labos Elemer_, May 24 2002

%E Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_