A210650 Least number k>0 such that at least one of (2^k-1)*2^n-1, (2^k-1)*2^n+1, (2^k+1)*2^n-1, or (2^k+1)*2^n+1 is a prime number.

1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 5, 5, 7, 2, 3, 4, 5, 3, 1, 1, 2, 3, 1, 5, 1, 4, 1, 2, 8, 1, 3, 1, 4, 3, 5, 5, 2, 17, 3, 4, 3, 21, 2, 1, 5, 8, 10, 7, 4, 1, 12, 3, 1, 3, 1, 3, 5, 8, 10, 8, 2, 4, 20, 2, 1, 11, 4, 14, 4, 3, 4, 17, 12, 2, 8, 13, 7, 1, 13, 5, 3, 5, 1, 8, 4, 13, 6, 3, 21, 10, 5, 1, 12, 7, 5, 1, 12, 3, 9, 7

Offset

1,9

Comments

More than 73% of a(n) are < n/10.

Conjecture: for n>1 a(n) is always < n.

Define partial sums S(N) = sum_{i=1..N} a(i) and T(N)=sum_{i=1..N} i = A000217(N). Then as N increases S(N)/T(n) tends to 0.078.

Links

Pierre CAMI, Table of n, a(n) for n = 1..10000

Maple

A210650 := proc(n)
        for k from 1 do
                if isprime((2^k-1)*2^n-1) then
                        return k;
                elif isprime((2^k-1)*2^n+1) then
                        return k;
                elif isprime((2^k+1)*2^n-1) then
                        return k;
                elif isprime((2^k+1)*2^n+1) then
                        return k;
                end if;
        end do:
end proc: # R. J. Mathar, Apr 02 2012

Mathematica

lnk[n_]:=Module[{k=1, t=2^n}, While[!AnyTrue[Flatten[{t(2^k-1)+{1, -1}, t(2^k+1)+ {1, -1}}], PrimeQ], k++]; k]; Array[lnk, 120] (* The program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 31 2015 *)

Prog

PFGW64 and SCRIPTIFY from Primeform group
Command: PFGW64 -f in.txt
in.txt file :
SCRIPT
DIM nn, 0
DIM kk
DIMS tt
OPENFILEOUT myfile, a(n).txt
LABEL loopn
SET nn, nn+1
IF nn>10000 THEN END
SET kk, 0
LABEL loopk
SET kk, kk+1
IF kk>2*nn THEN GOTO loopn
SETS tt, %d, %d, %d, %d\,; nn; kk; -1; -1
PRP 2^nn*(2^kk-1)-1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
SETS tt, %d, %d, %d, %d\,; nn; kk; -1; 1
PRP 2^nn*(2^kk-1)+1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
SETS tt, %d, %d, %d, %d\,; nn; kk; 1; -1
PRP 2^nn*(2^kk+1)-1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
SETS tt, %d, %d, %d, %d\,; nn; kk; 1; 1
PRP 2^nn*(2^kk+1)+1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
GOTO loopk
LABEL a
WRITE myfile, tt
GOTO loopn

Crossrefs

Sequence in context: A073356 A093032 A072115 * A218754 A079948 A257657

Adjacent sequences: A210647 A210648 A210649 * A210651 A210652 A210653

Keyword

nonn

Author

Pierre CAMI, Mar 27 2012

Status

approved