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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-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

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