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A235863 Exponent of the multiplicative group G_n:={x+iy: x^2+y^2==1 (mod n); 0 <= x,y < n} where i=sqrt(-1).

%I A235863 #75 Jun 12 2022 06:45:45
%S A235863 1,2,4,4,4,4,8,4,12,4,12,4,12,8,4,4,16,12,20,4,8,12,24,4,20,12,36,8,
%T A235863 28,4,32,8,12,16,8,12,36,20,12,4,40,8,44,12,12,24,48,4,56,20,16,12,52,
%U A235863 36,12,8,20,28,60,4,60,32,24,16,12,12,68,16,24,8,72
%N A235863 Exponent of the multiplicative group G_n:={x+iy: x^2+y^2==1 (mod n); 0 <= x,y < n} where i=sqrt(-1).
%C A235863 From _Jianing Song_, Nov 05 2019: (Start)
%C A235863 Exponent of the group G is the least e > 0 such that x^e = 1 for every x in G, where 1 is the identity element.
%C A235863 Also the exponent of O(2,Z_n) or SO(2,Z_n). O(2,Z_n) is the group of 2 X 2 matrices A over Z_n such that A*A^T = E = [1,0;0,1]; SO(2,Z_n) is the group of 2 X 2 matrices A over Z_n such that A*A^T = E = [1,0;0,1] and det(A) = 1. Note that G_n is isomorphic to SO(2,Z_n) by the mapping x+yi <-> [x,y;-y,x]. See A060698 for the group structure of SO(2,Z_n) and A182039 for the group structure of O(2,Z_n). (End)
%H A235863 Andrew Howroyd, <a href="/A235863/b235863.txt">Table of n, a(n) for n = 1..10000</a>
%H A235863 José María Grau, A. M. Oller-Marcén, Manuel Rodriguez and Daniel Sadornil, <a href="http://arxiv.org/abs/1401.4708">Fermat test with Gaussian base and Gaussian pseudoprimes</a>, arXiv:1401.4708 [math.NT], 2014.
%H A235863 José María Grau, A. M. Oller-Marcén, Manuel Rodriguez and Daniel Sadornil, <a href="http://dx.doi.org/10.1007/s10587-015-0221-2">Fermat test with Gaussian base and Gaussian pseudoprimes</a>, Czechoslovak Mathematical Journal 65(140), (2015) pp. 969-982.
%F A235863 a(2) = 2, a(4) = a(8) = a(16) = 4, a(2^e) = 2^(e-2) for e >= 5; a(p^e) = (p-1)*p^(e-1) if p == 1 (mod 4) and (p+1)*p^(e-1) if p == 1 (mod 4). - _Jianing Song_, Nov 05 2019
%F A235863 If gcd(n,m)=1 then a(nm) = lcm(a(n), a(m)).
%t A235863 fa=FactorInteger; lam[1]=1;lam[p_, s_] := Which[Mod[p, 4] == 3, p ^ (s - 1 ) (p + 1) , Mod[p, 4] == 1, p ^ (s - 1 ) (p - 1)  , s ≥ 5, 2 ^ (s - 2 ), s > 1, 4, s == 1, 2];lam[n_] := {aux = 1; Do[aux = LCM[aux, lam[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length[fa[n]]}]; aux}[[1]] ; Array[lam, 100]
%o A235863 (PARI) a(n)={my(f=factor(n)); lcm(vector(#f~, i, my([p,e]=f[i,]); if(p==2, 2^max(e-2, min(e,2)), p^(e-1)*if(p%4==1, p-1, p+1))))} \\ _Andrew Howroyd_, Aug 06 2018
%Y A235863                                    (Z/nZ)*    ------    G_n
%Y A235863 Order:                             A000010    ------  A060968.
%Y A235863 Exponent:                          A002322    ------  this sequence.
%Y A235863                                      n-1      ------  A201629.
%Y A235863 Carmichael/G-Carmichael numbers:   A002997    ------  A235865.
%Y A235863 Lehmer /G-Lehmer numbers:          unknown    ------  A235864.
%Y A235863 Cyclic/G-cyclic numbers:           A003277    ------  A235866.
%Y A235863 n such that the group is cyclic:   A033948    ------  A235868.
%Y A235863 Cf. A060698, A182039.
%K A235863 nonn
%O A235863 1,2
%A A235863 _José María Grau Ribas_, Jan 16 2014