A316566
Triangle read by rows: T(n,k) is the number of elements of the group GL(2, Z(n)) with order k, 1 <= k <= A316565(n).
Original entry on oeis.org
1, 1, 3, 2, 1, 13, 8, 6, 0, 8, 0, 12, 1, 27, 8, 36, 0, 24, 1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80, 1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1, 57, 170, 42, 0, 618, 48, 84, 0, 0, 0, 84
Offset: 1
Triangle begins:
1
1, 3, 2
1, 13, 8, 6, 0, 8, 0, 12
1, 27, 8, 36, 0, 24
1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80
1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24
...
-
MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
row(n)={my(L=List()); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(gcd(lift(matdet(M)), n)==1, my(t=MatOrder(M)); while(#L
A066907
Number of elements in GL(2,Z_n) x with x^2 == I mod n where I is the identity matrix.
Original entry on oeis.org
1, 4, 14, 28, 32, 56, 58, 176, 110, 128, 134, 392, 184, 232, 448, 608, 308, 440, 382, 896, 812, 536, 554, 2464, 752, 736, 974, 1624, 872, 1792, 994, 2336, 1876, 1232, 1856, 3080, 1408, 1528, 2576, 5632, 1724, 3248, 1894, 3752, 3520, 2216, 2258, 8512, 2746
Offset: 1
Sharon Sela (sharonsela(AT)hotmail.com), Jan 26 2002
-
f[p_, e_] := p^(2*e-1)*(p+1) + 2; f[2, e_] := 9*4^(e-1)+32; f[2, 1] = 4; f[2, 2] = 28; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)
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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) \\ Charles R Greathouse IV, May 29 2013
A316560
Number of cyclic subgroups of the group GL(2, Z(n)), counting conjugates as distinct.
Original entry on oeis.org
1, 5, 28, 62, 176, 148, 610, 696, 1252, 920, 2296, 1972, 4874, 3523, 6040, 6320, 8136, 7348, 14984, 13568, 22124, 11920, 17396, 23952, 29846, 28172, 38044, 47656, 47282, 32908, 75036, 53520, 71768, 42312, 145852, 99892, 123524, 88456, 187036, 179200, 152290
Offset: 1
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Concatenation([1], List([2..7], n->Sum( Filtered( ConjugacyClassesSubgroups( GL(2, Integers mod n)), x->IsCyclic( Representative(x))), Size)));
-
MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
a(n)={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(gcd(lift(matdet(M)), n)==1, 1/eulerphi(MatOrder(M)))))))}
Showing 1-3 of 3 results.
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