A214497 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, 9, 9, 6, 3, 93, 3, 54, 18, 96, 213, 297, 1206, 258, 312, 201, 261, 1206, 1158, 396, 1062, 216, 708, 762, 816, 678, 3579, 762, 831, 2106, 4734, 576, 333, 633, 213, 2766, 363, 2454, 1464, 2007, 4551, 3183, 1497, 4899, 198, 66, 9984, 2847, 276, 3051

Offset

1,2

Comments

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

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

Links

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

Maple

A214497 := 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:
seq(A214497(n), n=1..80) ; # R. J. Mathar, Jul 23 2012

Mathematica

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 *)

Prog

(PFGW64 and SCRIPTIFY)
SCRIPT
DIM nn, 0
DIM kk
DIM jj
DIMS tt
OPENFILEOUT myfile, a(n).txt
OPENFILEOUT myf, b(n).txt
LABEL loopn
SET nn, nn+1
SET jj, 0
IF nn>500 THEN END
SET kk, -1
LABEL loopk
SET kk, kk+1
SETS tt, %d, %d\,; nn; kk
PRP (3^nn-kk)*2^nn-1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
GOTO loopk
LABEL a
SET jj, jj+1
PRP (3^nn-kk)*2^nn+1, tt
IF ISPRP THEN GOTO d
IF ISPRIME THEN GOTO d
GOTO loopk
LABEL d
WRITE myfile, tt
SETS tt, %d, %d\,; nn; jj
WRITE myf, tt
GOTO loopn

Crossrefs

Cf. A214495-A214498.

Sequence in context: A180596 A154465 A176533 * A177015 A355971 A327000

Adjacent sequences: A214494 A214495 A214496 * A214498 A214499 A214500

Keyword

nonn

Author

Pierre CAMI, Jul 20 2012

Status

approved