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 A182039 Order of the group O(2,Z_n); number of orthogonal 2 X 2 matrices over the ring Z/nZ. 5
 1, 2, 8, 16, 8, 16, 16, 64, 24, 16, 24, 128, 24, 32, 64, 128, 32, 48, 40, 128, 128, 48, 48, 512, 40, 48, 72, 256, 56, 128, 64, 256, 192, 64, 128, 384, 72, 80, 192, 512, 80, 256, 88, 384, 192, 96, 96, 1024, 112, 80, 256, 384, 104, 144, 192, 1024, 320, 112, 120, 1024, 120, 128, 384, 512, 192, 384, 136, 512, 384, 256, 144, 1536 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of matrices M=[a,b;c,d] with 0<=a,b,c,d= 3, where the generators are [-1,0;0,1], [2^(n-1)+1,0;0,1] along with the generators of SO(2,Z_2^e) shown in A060968 if e >= 3. The exponent of O(2,Z_n) (i.e., least e > 0 such that x^e = E for every x in O(2,Z_n)) is given by A235863(n). The rank of O(2,Z_n) (i.e., the minimum number of generators) is 2*omega(n) if n is odd, 2*omega(n)-1 if n is even but not divisible by 4, 2*omega(n)+1 if n is divisible by 4 but not by 8 and 2*omega(n)+3 is n is divisible by 8, omega = A001221. The smallest n divisible by 8 such that rank(O(2,Z_n)) < rank(O(2,Z_(n+1))) is n = 1784: O(2,Z_1784) = C_2 X C_2 X C_2 X C_2 X C_2 X C_4 X C_224, while O(2,Z_1785) = C_2 X C_2 X C_2 X C_2 X C_4 X C_4 X C_8 X C_16. The smallest n divisible by 8 such that rank(O(2,Z_n)) < rank(O(2,Z_(n-1))) is n = 256. (End) LINKS Jianing Song, Table of n, a(n) for n = 1..10000 (first 1000 terms from Joerg Arndt) FORMULA From Jianing Song, Nov 05 2019: (Start) a(n) = A060968(n) * A060594(n). Multiplicative with a(2) = 1, a(4) = 16, a(2^e) = 2^(e+3) for e >= 3; a(p^e) = 2*(p-1)*p^(e-1) if p == 1 (mod 4), 2*(p+1)*p^(e-1) if p == 3 (mod 4). (End) EXAMPLE a(1) = 1 because 1 = 0 in the zero ring, so although there only exists the zero matrix, it still equals the unit matrix. O(2,Z_6) = {[0,1;5,0], [0,1;1,0], [0,5;1,0], [0,5;5,0], [1,0;0,1], [1,0;0,5], [2,3;3,2], [2,3;3,4], [3,2;4,3], [3,2;2,3], [3,4;2,3], [3,4;4,3], [4,3;3,4], [4,3;3,2], [5,0;0,5], [5,0;0,1]}, so a(6) = 16. For n = 16, SO(2,Z_16) is generated by [9,0;0,9], [0,1;-1,0], and [4,1;-1,4] (see Jianing Song link in A060968), so O(2,Z_16) is generated by [-1,0;0,1], [9,0;0,1], [9,0;0,9], [0,1;-1,0], and [4,1;-1,4], which gives O(2,Z_16) = C_2 X C_2 X C_2 X C_4 X C_4, so a(16) = 128. MATHEMATICA gg[n_]:=gg[n]=Flatten[Table[{{x, y}, {z, t}}, {x, n}, {y, n}, {t, n}, {z, n}], 3]; orto=1; orto[n_]:=orto[n]=Length@gg[n][[Select[Range[Length[gg[n]]], Mod[gg[n][[#]].Transpose[gg[n][[#]]], n]=={{1, 0}, {0, 1}}&]]]; Table[Print[orto[n]]; orto[n], {n, 1, 22}] PROG (PARI) a(n)= {     my(r=1, f=factor(n));     for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);         if(p==2 && e==1, r*=2);         if(p==2 && e==2, r*=16);         if(p==2 && e>=3, r*=2^(e+3));         if(p%4==1, r*=2*(p-1)*p^(e-1));         if(p%4==3, r*=2*(p+1)*p^(e-1));     );     return(r); } \\ Jianing Song, Nov 05 2019 CROSSREFS Cf. A060968 (order of SO(2,Z_n)), A060594, A235863, A001221, A209411. Sequence in context: A167592 A094513 A110004 * A174882 A080095 A193219 Adjacent sequences:  A182036 A182037 A182038 * A182040 A182041 A182042 KEYWORD nonn,mult AUTHOR José María Grau Ribas, Apr 07 2012 EXTENSIONS Terms beyond a(22) by Joerg Arndt, Apr 13 2012 a(1) changed to 1 by Andrey Zabolotskiy, Nov 13 2019 STATUS approved

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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)