A316553 Number of elements of order 2 in the group SL(2, Z(n)).
0, 3, 1, 7, 1, 7, 1, 15, 1, 7, 1, 15, 1, 7, 3, 15, 1, 7, 1, 15, 3, 7, 1, 31, 1, 7, 1, 15, 1, 15, 1, 15, 3, 7, 3, 15, 1, 7, 3, 31, 1, 15, 1, 15, 3, 7, 1, 31, 1, 7, 3, 15, 1, 7, 3, 31, 3, 7, 1, 31, 1, 7, 3, 15, 3, 15, 1, 15, 3, 15, 1, 31, 1, 7, 3, 15, 3, 15, 1
Offset: 1
Keywords
Examples
Case n=2: the three 2 X 2 matrices on Z(2) having determinant 1 and order 2 are: [ 0 1 ] [ 1 0 ] [ 1 1 ] [ 1 0 ] [ 1 1 ] [ 0 1 ]
Links
- Antti Karttunen, Table of n, a(n) for n = 1..1023
Programs
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GAP
Concatenation([0], List([2..10], n->Sum(Filtered( ConjugacyClassesSubgroups( SL(2, Integers mod n)), x->Order( Representative(x))=2 and IsCyclic( Representative(x))), Size)));
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PARI
a(n)={my(id=matid(2)); sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, M^2==id))))) - 1} -
PARI
memoA316553 = Map(); \\ Only values at 2^k are actually collected here. A316553slow_memoized(n) = if(1==n, 0, if((n%2)&&isprimepower(n), 1, my(id=matid(2), v); if(mapisdefined(memoA316553,n,&v), v, v = (sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, M^2==id))))) - 1); mapput(memoA316553,n,v); (v)))); A316553(n) = if(1==n,0,my(f=factor(n)); -1 + prod(i=1,#f~,1+A316553slow_memoized(f[i,1]^f[i,2]))); \\ (Based on Robert Israel's multiplicativity rule) - Antti Karttunen, Dec 05 2021
Formula
Conjecture: a(n) = 2^(omega(n) + min(3, valuation(n, 2))) - 1.
From Robert Israel, Jun 15 2020: (Start)
Number of solutions mod n, other than t[1]=t[4]=1,t[2]=t[3]=0, of the equations t[2]*(t[1] + t[4])=0, t[3]*(t[1] + t[4])=0, t[1]^2 + t[2]*t[3] = 1, t[2]*t[3] + t[4]^2 = 1, t[1]*t[4] - t[2]*t[3] = 1.
If m and n are coprime, a(m*n) = a(m)*a(n)+a(m)+a(n) (i.e. a(n)+1 is multiplicative).
If n > 1 is in A061345, a(n)=1. (End)
Comments