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A298940 a(n) is the smallest positive integer k such that 3^n - 2 divides 3^(n + k) + 2, or 0 if there is no such k. 1
1, 3, 10, 39, 60, 121, 0, 117, 4920, 0, 0, 0, 28322, 0, 1434890, 0, 0, 0, 116226146, 0, 0, 15690529803, 0, 108443565, 66891206007, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22514195294549868, 0, 405255515301897626, 0, 1823649818858539320, 0, 0, 5861731560616733529, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

3^n - 2 divides 3^(n + (2m + 1) * a(n)) + 2 for all nonnegative integers m.

a(n) is the least positive integer k, if any, such that 3^k == -1 (mod 3^n-2). If the order of 3 mod p is odd for some prime p dividing 3^n-2, a(n)=0. - Robert Israel, Feb 05 2018

LINKS

Robert Israel, Table of n, a(n) for n = 1..166

EXAMPLE

a(2) = 3 because 3^2 - 2 divides 3^5 + 2 and 3^2 - 2 does not divide any 3^x - 2 for 2 < x < 5.

a(5) = 60 because 3^5 - 2 divides 3^65 + 2 and 3^5 - 2 does not divide any 3^x - 2 for 5 < x < 65.

MAPLE

# This requires Maple 2016 or later

f:= proc(n) local m, ps, a, p, q, phiq, v, br, ar;

  m:= 3^n-2;

   ps:= ifactors(m)[2];

   a:= 0;

   for p in ps do

     q:= p[1]^p[2];

     phiq:= (p[1]-1)*p[1]^(p[2]-1);

     v:= NumberTheory:-MultiplicativeOrder(3, q);

     if v::odd then return 0 fi;

     if p[2]=1 then br:= v/2

     else br:= traperror(NumberTheory:-ModularLog(-1, 3, q));

          if br = lasterror then return 0 fi;

     fi;

     if a = 0 then a:= v; ar:= br

     else

        ar:= NumberTheory:-ChineseRemainder([ar, br], [a, v]);

        if ar = FAIL then return 0 fi;

        a:= ilcm(a, v);

     fi

   od:

   ar;

end proc:

f(1):= 1:

map(f, [$1..50]); # Robert Israel, Feb 06 2018

MATHEMATICA

a[1] = 1; a[n_] := If[IntegerQ[order = MultiplicativeOrder[3, 3^n - 2, {-1}]], order, 0]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 20}] (* Jean-Fran├žois Alcover, Feb 06 2018, after Robert Israel *)

PROG

(Python)

from sympy import discrete_log

def A298940(n):

    if n == 1:

        return 1

    try:

        return discrete_log(3**n-2, -1, 3)

    except ValueError:

        return 0 # Chai Wah Wu, Feb 05 2018

(PARI) a(n) = if(n==1, return(1)); my(l = znlog(-1, Mod(3, 3^n - 2))); if(l == [], return(0), return(l)) \\ Iain Fox, Feb 06 2018

CROSSREFS

Cf. A168607, A298827.

Sequence in context: A140710 A103296 A259859 * A327847 A111749 A149048

Adjacent sequences:  A298937 A298938 A298939 * A298941 A298942 A298943

KEYWORD

nonn

AUTHOR

Luke W. Richards, Jan 29 2018

EXTENSIONS

Corrected by Robert Israel, Feb 05 2018

STATUS

approved

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Last modified February 23 13:14 EST 2020. Contains 332159 sequences. (Running on oeis4.)