A066947 Number of elements of order 2 in GL(2,Z_n).
3, 13, 27, 31, 55, 57, 175, 109, 127, 133, 391, 183, 231, 447, 607, 307, 439, 381, 895, 811, 535, 553, 2463, 751, 735, 973, 1623, 871, 1791, 993, 2335, 1875, 1231, 1855, 3079, 1407, 1527, 2575, 5631, 1723, 3247, 1893, 3751, 3519, 2215, 2257, 8511, 2745
Offset: 2
Examples
a(3000) = (a(8)+1)*(a(3)+1)*(a(125)+1)-1 = (9*4^2 + 2)*(3^2 + 3 + 2)*(5^6 + 5^5 + 2) - 1 = 46204927 because 3000 = 2^3*3*5^3.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
- Alec Mihailovs, Problem 16 Solution
- Alec Mihailovs, Abstract Algebra with Maple
- Alec Mihailovs, Chapter 5. Cyclic Groups
Programs
-
Maple
Ord2inGL2 := proc(n::posint) local i,j,m,c; if n=1 then return 0 end if; m := ifactors(n)[2]; c := 1; j := 1; if (m[1,1]=2) then j := 2; if m[1,2]=1 then c := 4 elif m[1,2]=2 then c := 28 else c := 9*4^(m[1,2]-1)+32 end if end if; c := c*mul((m[i,1]+1)*m[i,1]^(2*m[i,2]-1)+2,i=j..nops(m))-1 end;
-
Mathematica
a[n_] := (c[0] = 1; c[1] = 4; c[2] = 28; c[k_] := 9*4^(k-1) + 32; fi = FactorInteger[n]; m = (s = Cases[fi, {2, _}]; If[s == {}, 0, s[[1, 2]]]); p = If[m == 0, fi, Rest[fi]]; p1 = p[[All, 1]]; p2 = p[[All, 2]]; c[m]*Times @@ (p1^(2p2) + p1^(2p2-1) + 2) - 1); (* Jean-François Alcover, May 19 2011, after Maple prog. *) -
PARI
a(n)=my(o=valuation(n,2),f=factor(n>>o)); prod(i=1,#f[,1],f[i,1]^(2*f[i,2])+f[i,1]^(2*f[i,2]-1)+2)*if(o, if(o>1, if(o>2, 9*4^(o-1)+32,28),4),1)-1 \\ Charles R Greathouse IV, May 29 2013
Formula
If n = 2^m*p^a...q^b where p, ..., q are the odd prime divisors of n, then a(n) = c(m)*(p^{2a} + p^{2a-1} + 2)...(q^{2b} + q^{2b-1} + 2) - 1 where c(0) = 1, c(1) = 4, c(2) = 28 and c(m) = 9*4^{m-1} + 32 for m > 2. The integer function f(n) = a(n)+1 is multiplicative, i.e., f(m*n)=f(m)*f(n) for coprime m and n. - Alec Mihailovs (alec(AT)mihailovs.com), Mar 24 2002
a(n) = A066907(n) - 1. - Andrew Howroyd, Jul 08 2018
a(n) = A227867(n) - 1 for odd n. - Jianing Song, Jul 08 2018