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%I #46 Sep 30 2020 12:21:03
%S 1,1,2,4,6,2,4,5,3,6,4,16,18,4,5,11,20,3,6,28,30,8,16,12,18,18,4,8,42,
%T 10,11,23,42,20,6,52,20,6,28,29,10,30,16,12,22,16,12,35,12,18,18,30,
%U 78,4,8,41,16,42,10,88,6,22,23,36,48,42,20,100,34,6,52,53,27,20,12,112,44
%N Multiplicative order of 3 mod m, where gcd(m, 3) = 1.
%C Essentially the same as A050975. - _R. J. Mathar_, Oct 13 2008
%C Let k, m be any positive numbers not divisible by 3. Let k <+> m denote that of the two numbers k + m, k + 2*m which is divisible by 3. Finally, for a number t divisible by 3, let t* = t/3^s where s is the 3-adic order of t. Let u = u(n) be the n-th number which is not divisible by 3. Consider the following algorithm of the calculating a(n), similar to the algorithm in A002326: Compute successively r_1 = (1 <+> u)*, r_2 = (r_1 <+> u)*, ..., r_h = (r_(h-1) <+> u)* and finish as soon as r_h = 1. Then a(n) = s(1 <+> u) + s(r_1 <+> u) + ... + s(r_(h-1) <+> u). Note that by a similar algorithm one can compute an arbitrary multiplicative order of a mod b, where gcd(a, b) = 1. - _Vladimir Shevelev_, Oct 06 2017
%H Antti Karttunen, <a href="/A053446/b053446.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = multiplicative order of 3 modulo floor((3*n-1)/2) = A001651(n), for n >= 1. - _Wolfdieter Lang_, Sep 28 2020
%e From _Vladimir Shevelev_, Oct 06 2017: (Start)
%e 7 is the fifth number not divisible by 3. According to the algorithm in the comment we have in the form of a "finite continued fraction"
%e 1 + 14
%e ------ + 7
%e 3
%e ---------- + 14
%e 3
%e ----------------- + 7
%e 3^2
%e ---------------------- = 1
%e 3^2
%e Summing the exponents of 3 in the denominators, we obtain a(5) = 1 + 1 + 2 + 2 = 6. (End)
%t MultiplicativeOrder[3, #] & /@ Select[ Range@ 115, GCD[3, #] == 1 &] (* _Robert G. Wilson v_, Apr 05 2011 *)
%o (PARI) lista(nn) = {for (n=1, nn, if (gcd(n,3) == 1, print1(znorder(Mod(3, n)), ", ")););} \\ _Michel Marcus_, Feb 06 2015
%o (Sage)
%o [Mod(3,n).multiplicative_order() for n in (1..115) if gcd(n,3) == 1] # _Peter Luschny_, Oct 07 2017
%o (GAP) List(Filtered([1..130],n->Gcd(n,3)=1),n->OrderMod(3,n)); # _Muniru A Asiru_, Feb 26 2019
%Y Cf. A001651, A002326, A050975.
%K nonn
%O 1,3
%A _David W. Wilson_
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