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

1,2

Comments

Equivalently, the number of cyclic subgroups of the group SL(2, Z(n)) having order 2, counting conjugates as distinct.

Links

Antti Karttunen, Table of n, a(n) for n = 1..1023

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)

Example

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 ]

Prog

(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)));
(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

Crossrefs

Column 2 of A316564.

Cf. A174275, A316537.

Cf. A061345.

Sequence in context: A038870 A347234 A324865 * A347959 A347963 A186428

Adjacent sequences: A316550 A316551 A316552 * A316554 A316555 A316556

Keyword

nonn

Author

Andrew Howroyd, Jul 06 2018

Status

approved