|
%I #51 Feb 07 2018 02:43:10
%S 1,3,10,39,60,121,0,117,4920,0,0,0,28322,0,1434890,0,0,0,116226146,0,
%T 0,15690529803,0,108443565,66891206007,0,0,0,0,0,0,0,0,0,0,0,
%U 22514195294549868,0,405255515301897626,0,1823649818858539320,0,0,5861731560616733529,0,0,0
%N 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.
%C 3^n - 2 divides 3^(n + (2m + 1) * a(n)) + 2 for all nonnegative integers m.
%C 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
%H Robert Israel, <a href="/A298940/b298940.txt">Table of n, a(n) for n = 1..166</a>
%e 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.
%e 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.
%p # This requires Maple 2016 or later
%p f:= proc(n) local m,ps,a,p,q,phiq,v,br,ar;
%p m:= 3^n-2;
%p ps:= ifactors(m)[2];
%p a:= 0;
%p for p in ps do
%p q:= p[1]^p[2];
%p phiq:= (p[1]-1)*p[1]^(p[2]-1);
%p v:= NumberTheory:-MultiplicativeOrder(3,q);
%p if v::odd then return 0 fi;
%p if p[2]=1 then br:= v/2
%p else br:= traperror(NumberTheory:-ModularLog(-1,3,q));
%p if br = lasterror then return 0 fi;
%p fi;
%p if a = 0 then a:= v; ar:= br
%p else
%p ar:= NumberTheory:-ChineseRemainder([ar,br],[a,v]);
%p if ar = FAIL then return 0 fi;
%p a:= ilcm(a, v);
%p fi
%p od:
%p ar;
%p end proc:
%p f(1):= 1:
%p map(f, [$1..50]); # _Robert Israel_, Feb 06 2018
%t 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_ *)
%o (Python)
%o from sympy import discrete_log
%o def A298940(n):
%o if n == 1:
%o return 1
%o try:
%o return discrete_log(3**n-2,-1,3)
%o except ValueError:
%o return 0 # _Chai Wah Wu_, Feb 05 2018
%o (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
%Y Cf. A168607, A298827.
%K nonn
%O 1,2
%A _Luke W. Richards_, Jan 29 2018
%E Corrected by _Robert Israel_, Feb 05 2018
|
