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

1,2

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

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 *)

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, 0
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-A214497.

Sequence in context: A329583 A050132 A320038 * A128756 A345056 A049784

Adjacent sequences: A214495 A214496 A214497 * A214499 A214500 A214501

Keyword

nonn

Author

Pierre CAMI, Jul 20 2012

Status

approved