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%I #38 Feb 26 2024 13:58:30
%S 1,1,2,3,3,5,6,2,4,9,3,11,10,9,14,5,5,6,18,6,10,7,6,23,21,4,26,10,9,
%T 29,30,3,6,33,11,35,9,10,15,39,27,41,4,14,11,6,5,18,24,15,50,51,6,53,
%U 18,18,14,22,6,12,55,10,50,7,7,65,9,18,34,69,23,30,14,21,74,15,12,10,26
%N Multiplicative order of 4 mod 2n+1.
%C For a set S = {x, y} (x < y), let f(S) = {2x, y - x}, then a(n) is the smallest k > 0 such that f_k({1, 2n}) = {1, 2n} where f_k(S) denotes iteration for k times. E.g., for n = 3 we have: f_1({1, 6}) = f({1, 6}) = {2, 5}, f_2({1, 6}) = f({2, 5}) = {3, 4}, f_3({1, 6}) = f({3, 4}) = {1, 6}. - _Jianing Song_, Jan 27 2019
%C From _Jianing Song_, Dec 24 2022: (Start)
%C Let psi = A002322. For n > 0, we have 4^(psi(2*n+1)/2) = 2^psi(2*n+1) == 1 (mod 2*n+1), so a(n) divides psi(2*n+1)/2 => a(n) <= psi(2*n+1)/2 <= n. a(n) = psi(2*n+1)/2 if and only if one of the two following conditions holds: (a) the multiplicative order of 2 modulo 2*n+1 is psi(2*n+1); (b) the multiplicative order of 2 modulo 2*n+1 is psi(2*n+1)/2, and psi(2*n+1) == 2 (mod 4).
%C Additionally, a(n) = n if and only if 2*n+1 = p is a prime, and one of the two following conditions holds: (a) 2 is a primitive root modulo p; (b) p == 3 (mod 4), and the multiplicative order of 2 modulo p is (p-1)/2 (in this case, we have p == 7 (mod 8) since 2 is a quadratic residue modulo p). Such primes p are listed in A216371. (End)
%H Vincenzo Librandi, <a href="/A053447/b053447.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiplicativeOrder.html">Multiplicative Order</a>
%F Let b = A002326, then a(n) = b(n) if b(n) is odd, otherwise a(n) = b(n)/2. - _Joerg Arndt_, Feb 03 2019
%t Table[ MultiplicativeOrder[4, n], {n, 1, 160, 2}] (* _Robert G. Wilson v_, Apr 05 2011 *)
%o (Magma) [1] cat [Modorder(4, 2*n+1): n in [1..100]]; // _Vincenzo Librandi_, Apr 01 2014
%o (PARI) a(n) = znorder(Mod(4, 2*n+1)); \\ _Michel Marcus_, Feb 05 2015
%o (GAP) List([0..80],n->OrderMod(4,2*n+1)); # _Muniru A Asiru_, Feb 25 2019
%Y Cf. A050976, A070667-A070675, A002326, A070676, A070677, A070681, A070678, A053451, A070679, A070682, A070680, A070683, A302141.
%K nonn
%O 0,3
%A _David W. Wilson_
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