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, 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

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-François 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