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

Author

Luke W. Richards, Jan 29 2018

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A168607, A298827.

Programs

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 *)

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

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

Extensions

Corrected by Robert Israel, Feb 05 2018