A214498 Smallest k>=0 such that (3^n+k)*2^n-1 and (3^n+k)*2^n+1 are a twin prime pair.

0, 6, 3, 12, 78, 18, 18, 141, 18, 54, 78, 132, 138, 78, 57, 537, 237, 6, 972, 219, 81, 3369, 69, 501, 2328, 18, 738, 291, 393, 969, 324, 492, 102, 3291, 1788, 1401, 891, 954, 4017, 309, 702, 1656, 3999, 1014, 2346, 4008, 3, 5001, 2736, 558, 2262, 969, 762

Offset: 1

Author

Pierre CAMI, Jul 20 2012

Keywords

• nonn

Comments

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

Heuristically, the average of a(n)/n^2 over n=1 to N tends to 1.2 as N increases.

Links

• Pierre CAMI, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A214495-A214497.

Programs

Maple

A214498 := proc(n)
    local k;
    for k from 0 do
        p := (3^n+k)*2^n-1 ;
        if isprime(p) and isprime(p+2) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 23 2012

Mathematica

sk[n_]:=Module[{n3=3^n,n2=2^n,k=0},While[!And@@PrimeQ[(n3+k)n2+{1,-1}], k++];k]; Array[sk,60] (* Harvey P. Dale, Jul 23 2013 *)