A000056 -id:A000056 - cp's OEIS frontend

A316537 Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

1, 5, 13, 28, 49, 73, 116, 176, 202, 265, 378, 464, 550, 636, 842, 936, 1041, 1183, 1486, 1712, 2082, 2055, 2120, 3088, 2114, 3023, 2503, 4200, 4238, 4862, 4902, 4648, 6564, 5749, 7434, 7688, 6331, 8190, 9880, 11344, 10172, 12066, 9378, 13224, 14168, 11612

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

			Case n=2: generators of the 5 cyclic groups are:
  [ 1 0 ]   [0 1]   [1 0]   [1 1]   [0 1]
  [ 0 1 ]   [1 0]   [1 1]   [0 1]   [1 1]

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000056, A062314, A316536, A316553, A316560, A316563, A316564.

Concatenation([1], List([2..10], n->Sum( Filtered( ConjugacyClassesSubgroups( SL(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(matdet(M)==1, 1/eulerphi(MatOrder(M)))))))}

a(n) = Sum_{k=1..A316563(n)} 1/phi(A316564(n, k)).

A316563 Maximum order of an element in the special linear group SL(2, Z(n)).

Original entry on oeis.org

1, 3, 6, 6, 10, 12, 14, 8, 18, 30, 22, 12, 26, 42, 30, 16, 34, 18, 38, 30, 42, 66, 46, 24, 50, 78, 54, 42, 58, 60, 62, 32, 66, 102, 70, 36, 74, 114, 78, 40, 82, 84, 86, 66, 90, 138, 94, 48, 98, 150, 102, 78, 106, 54, 110, 56, 114, 174, 118, 60, 122, 186, 126

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Row lengths of A316564.

Cf. A000056, A316537, A316565.

Concatenation([1], List([2..15], n->Maximum(List(SL(2, Integers mod n), Order))));

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
a(n)={my(m=0); 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(matdet(M)==1, m=max(m, MatOrder(M))))))); m}

A329119 Orders of the finite groups SL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 24, 60, 120, 336, 504, 720, 1320, 2184, 4080, 4896, 6840, 12144, 15600, 19656, 24360, 29760, 32736, 50616, 68880, 79464, 103776, 117600, 148824, 205320, 226920, 262080, 300696, 357840, 388944, 492960, 531360, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920

Offset: 1

Jianing Song, Nov 05 2019

nonn

SL_2(K) means the group of 2 X 2 matrices A over K such that det(A) = 1.
In general, let R be any commutative ring with unity, GL_n(R) be the group of n X n matrices A over R such that det(A) != 0 and SL_n(R) be the group of n X n matrices A over R such that det(A) = 1, then GL_n(R)/SL_n(R) = R* is the multiplicative group of R. This is because if we define f(M) = det(M) for M in GL_n(R), then f is a surjective homomorphism from GL_n(K) to R*, and SL_n(R) is its kernel. Thus |GL_n(R)|/|SL_n(R)| = |R*|; if K is a finite field, then |GL_n(R)|/|SL_n(R)| = |K|-1.
Also a(n) is the order of PGL_2(K) when K is a finite field with q = A246655(n) elements. Note that PGL(m,q) and SL(m,q) are not isomorphic unless gcd(m,q-1) = 1. For example, PGL(2,3) = S_4 is not isomorphic to SL(2,3), PGL(2,5) = S_5 is not isomorphic to SL(2,5). - Jianing Song, Apr 04 2022

			a(4) = 120 because A246655(4) = 5, and 5*(5^2-1) = 120.

Robert Israel, Table of n, a(n) for n = 1..10000
Groupprops, Projective general linear group of degree two
Groupprops, Special linear group of degree two

Subsequence of A007531.
Cf. A246655, A000056 (order of SL_2(Z_n)).
For the order of GL_2(K) see A059238.

N:= 200:
P:= select(isprime, {2,seq(i,i=3..N,2)}):
PP:= map(proc(p) local i; seq(p^i,i=1..floor(log[p](N))) end proc, P):
map(t -> t*(t^2-1), sort(convert(PP,list))); # Robert Israel, Nov 13 2019

p = Select[Range[200], PrimePowerQ];
(p-1) p (p+1) (* Jean-François Alcover, Aug 22 2020 *)

[(p+1)*p*(p-1) | p <- [1..200], isprimepower(p)]

If the finite field K has q elements, then the order of the group SL_2(K) is q*(q^2-1).

a(n) = A059238(n)/(A246655(n)-1) = A007531(A246655(n)+1).

A001766 Index of (the image of) the modular group Gamma(n) in PSL_2(Z).

Original entry on oeis.org

1, 6, 12, 24, 60, 72, 168, 192, 324, 360, 660, 576, 1092, 1008, 1440, 1536, 2448, 1944, 3420, 2880, 4032, 3960, 6072, 4608, 7500, 6552, 8748, 8064, 12180, 8640, 14880, 12288, 15840, 14688, 20160, 15552, 25308, 20520, 26208, 23040, 34440, 24192, 39732, 31680

Offset: 1

N. J. A. Sloane

Equivalently, the degree of the modular curve X(N) as a cover of the j-line.

R. C. Gunning, Lectures on Modular Forms, Princeton Univ. Press, Princeton, NJ, 1962, p. 15.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ioannis Ivrissimtzis, David Singerman, and James Strudwick, From Farey fractions to the Klein quartic and beyond, arXiv:1909.08568 [math.GR], 2019. See mu(n), p. 3.
Index entries for sequences related to modular groups.

Cf. A000056, A000114, A002117.

proc(n) local b,d: b := (n^3)/2: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1-d^(-2)): fi: od: RETURN(b): end:

Table[ (n^3)/If[ n>2, 2, 1 ] Times@@(1-1/Select[ Range[ n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]^2), {n, 1, 45} ] (* Olivier Gérard, Aug 15 1997 *)

a(n) = if (n==1, 1, if (n==2, 6, my(f=factor(n)); prod(k=1, #f~, 1-1/f[k,1]^2)*n^3/2)); \\ Michel Marcus, Oct 23 2019

a(n) = A000056(n) for n = 2 and (1/2)*A000056(n) for n > 2 (since -I is contained in Gamma(2) but not in Gamma(n) for n > 2).
a(n) = n * A000114(n). - Michael Somos, Jan 29 2004
a(n) = ((n^3)/2)*Product_{p | n, p prime} (1-1/p^2), for n>=3. - Michel Marcus, Oct 23 2019
Sum_{k=1..n} a(k) ~ n^4 / (8*zeta(3)). - Amiram Eldar, Jun 01 2025

More terms from Olivier Gérard, Aug 15 1997

Definition corrected by Mira Bernstein, May 30 2006

A115224 Number of 3 X 3 symmetric matrices over Z(n) having determinant 1.

Original entry on oeis.org

1, 28, 234, 896, 3100, 6552, 16758, 28672, 56862, 86800, 160930, 209664, 371124, 469224, 725400, 917504, 1419568, 1592136, 2475738, 2777600, 3921372, 4506040, 6435814, 6709248, 9687500, 10391472, 13817466, 15015168, 20510308, 20311200, 28628190, 29360128, 37657620

Offset: 1

T. D. Noe, Jan 16 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..4885 from Enrique Pérez Herrero)

Cf. A000056 (order of the group SL(2, Z_n)), A011785 (number of 3 X 3 matrices whose determinant is 1 mod n, i.e. order of SL(3, Z_n)).

Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]==1, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}]; cnt, {n, 2, 20}]
JordanTotient[n_,k_:1] := DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A115224[n_IntegerQ] := JordanTotient[n^2,3]/n; Table[A115224[n], {n,100}] (* Enrique Pérez Herrero, Sep 14 2010 *)
f[p_, e_] := (p^3 - 1)*p^(5*e - 3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 15 2020 *)

a(1)=1 because the matrix of all zeros has determinant 0, but 0=1 (mod 1).
For prime p, a(p) = (p^3-1)*p^2.
Multiplicative with a(p^e) = (p^3-1)*p^(5e-3).
a(n) = A011785(n)/A000056(n).
a(n) = A059376(n^2)/n. - Enrique Pérez Herrero, Sep 14 2010
a(n) = n^2*A059376(n). Dirichlet g.f.: zeta(s-5)/zeta(s-2). - R. J. Mathar, Feb 27 2011
Sum_{k=1..n} a(k) ~ 15*n^6 / Pi^4. - Vaclav Kotesovec, Feb 07 2019
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/(1 - p^3 - p^5 + p^8)) = 1.04172462829914219180789244796430293454403616906393417764614215669994022537... - Vaclav Kotesovec, Sep 20 2020

A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 24, 168, 1, 1, 1, 48, 5616, 20160, 1, 1, 1, 120, 43008, 12130560, 9999360, 1, 1, 1, 144, 372000, 660602880, 237783237120, 20158709760, 1, 1, 1, 336, 943488, 29016000000, 167761422581760, 42064805779476480, 163849992929280, 1

Offset: 0

Andrew Howroyd, Jul 08 2018

All rows are multiplicative.
Equivalently, the number of n X n matrices mod k with determinant 1.
Also, for k prime (but not higher prime powers) the number of n X n matrices over GF(k) with determinant 1.

			Array begins:
==============================================================
n\k| 1       2        3         4           5           6
---+----------------------------------------------------------
0  | 1       1        1         1           1            1 ...
1  | 1       1        1         1           1            1 ...
2  | 1       6       24        48         120          144 ...
3  | 1     168     5616     43008      372000       943488 ...
4  | 1   20160 12130560 660602880 29016000000 244552089600 ...
5  | 1 9999360 ...
...

R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Mathematics 66 (1987) pp. 35-49.
J. M. Lockhart and W. P. Wardlaw, Determinants of Matrices over the Integers Modulo m, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.
The Group Properties Wiki, Order formulas for linear groups

Rows n=2..4 are A000056, A011785, A011786.
Columns k=2..5, 7 are A002884, A003787, A011787, A003789, A003790.
Cf. A316622.

T:=function(n,k) if k=1 or n=0 then return 1; else return Order(SL(n, Integers mod k)); fi; end;
for n in [0..5] do Print(List([1..6], k->T(n,k)), "\n"); od;

T[n_, k_] := If[k == 1 || n == 0, 1, k^(n^2-1) Product[1 - p^-j, {p, FactorInteger[k][[All, 1]]}, {j, 2, n}]];
Table[T[n-k+1, k], {n, 0, 8}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Sep 19 2019 *)

T(n,k)={my(f=factor(k)); if(n<1, n==0, k^(n^2-1) * prod(i=1, #f~, my(p=f[i,1]); prod(j=2, n, (1 - p^(-j)))))}

T(n,p^e) = (p^e)^(n^2-1) * Product_{j=2..n} (1 - 1/p^j) for prime p, n > 0.

A070732 Size of largest conjugacy class in the group GL(2,Z_n).

Original entry on oeis.org

1, 3, 12, 12, 30, 36, 56, 48, 108, 90, 132, 144, 182, 168, 360, 192, 306, 324, 380, 360, 672, 396, 552, 576, 750, 546, 972, 672, 870, 1080, 992, 768, 1584, 918, 1680, 1296, 1406, 1140, 2184, 1440, 1722, 2016, 1892, 1584, 3240, 1656, 2256, 2304, 2744, 2250

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A062354.
Cf. A000010, A000056.
Cf. A000082, A099892, A098198, A330523.

f[n_] := Block[{a = 1, b = FactorInteger[n]}, While[ Length[b] > 0, a = a*(b[[1, 1]] + 1)*b[[1, 1]]^(2b[[1, 2]] - If[ OddQ[ b[[1, 1]]], 1, 2]); b = Drop[b, 1]]; a]; Table[ f[n], {n, 1, 55}]
Table[n*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]/EulerPhi[2*n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
f[p_, e_] := (p + 1)*p^(2*e - If[p == 2, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]+1)*f[i,1]^(2*f[i,2] - if(f[i,1]==2,2,1)));} \\ Amiram Eldar, Nov 05 2022

Multiplicative with a(p^e) = (p+1)*p^(2e - k), k = 1 if p is odd, k = 2 if p is 2.
a(n) = A000056(n)/A000010(2*n). - Vladeta Jovovic, Dec 22 2003
From R. J. Mathar, Apr 14 2011: (Start)
Dirichlet g.f.: (2^s-1)*zeta(s-1)*zeta(s-2)/((2^s+2)*zeta(2s-2)).
Dirichlet convolution of A000082 with a signed variant of A099892. (End)
Sum_{k=1..n} a(k) ~ 7*n^3 / (2*Pi^2). - Vaclav Kotesovec, Feb 01 2019
Sum_{n>=1} 1/a(n) = (13/11) * zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = (13/11) * A098198 * A330523 = 1.7136743536... . - Amiram Eldar, Nov 05 2022

Edited by Robert G. Wilson v, May 20 2002

A115076 Number of 2 X 2 symmetric matrices over Z(n) having determinant 1.

Original entry on oeis.org

1, 4, 6, 12, 30, 24, 42, 48, 54, 120, 110, 72, 182, 168, 180, 192, 306, 216, 342, 360, 252, 440, 506, 288, 750, 728, 486, 504, 870, 720, 930, 768, 660, 1224, 1260, 648, 1406, 1368, 1092, 1440, 1722, 1008, 1806, 1320, 1620, 2024, 2162, 1152, 2058, 3000

Offset: 1

T. D. Noe, Jan 12 2006

a(1)=1 because the matrix of all zeros has determinant 0, but 0=1 (mod 1).

Andrew Howroyd, Table of n, a(n) for n = 1..2500

Cf. A000056 (order of the group SL(2, Z_n)), A175647, A243380.

Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]==1, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 50}]
f[p_, e_] := If[Mod[p, 4] == 1, (p+1)*p^(2*e-1), (p-1)*p^(2*e-1)]; f[2, 1] = 4; f[2, e_] := 3*2^(2*e-2); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 28 2023 *)

a(n)={my(v=vector(n)); for(i=0, n-1, for(j=0, n-1, v[i*j%n+1]++)); sum(i=0, n-1, v[(i^2+1)%n+1])} \\ Andrew Howroyd, Jul 04 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); p^(2*e-1)*if(p==2, if(e==1, 2, 3/2), if(p%4==1, p+1, p-1)))} \\ Andrew Howroyd, Jul 04 2018

Multiplicative with a(2^1) = 4, a(2^e) = 3*2^(2*e-2) for e > 1, a(p^e) = (p+1)*p^(2*e-1) for p mod 4 == 1, a(p^e) = (p-1)*p^(2*e-1) for p mod 4 == 3. - Andrew Howroyd, Jul 04 2018

Sum_{k=1..n} a(k) ~ c * n^3, where c = (5/(2*Pi^2)) * A175647 * A243380 = 0.282098596071... . - Amiram Eldar, Aug 28 2023

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Original entry on oeis.org

1, 10, 6, 33, 24, 24, 88, 48, 72, 48, 145, 120, 120, 120, 120, 330, 144, 240, 198, 240, 144, 385, 336, 336, 336, 336, 336, 336, 736, 384, 576, 384, 672, 384, 576, 384, 945, 648, 648, 864, 648, 648, 864, 648, 648, 1450, 720, 1200, 720, 1200, 870, 1200, 720, 1200, 720

Offset: 1

Erdos Pal, Oct 03 2010

The n-th row is {T(n,0),T(n,1),...,T(n,n-1)}.
Let m denote the prime power p^e.
T(m,0) = A020478(m) = (p^(e+1) + p^e-1)*p^(2*e-1).
T(m,1) = A000056(m) = (p^2-1)*p^(3*e-2).
T(prime(n),1) = A127917(n).
Sum_{k=1..n-1} T(n,k) = A005353(n).
T(n,1) = n*A007434(n) for n>=1 because A000056(n) = n*Jordan_Function_J_2(n).
T(2^n,1) = A083233(n) = A164640(2n) for n>=1. Proof: a(n):=T(2^n,1); a(1)=6, a(n)=8*a(n-1); A083233(1)=6 and A083233(n) is a geometric series with ratio 8 (because of its g.f.), too; A164640 = {b(1)=1, b(2)=6, b(n)=8*b(n-2)}.
T(2^n,0) = A165148(n) for n>=0, because  2*T(2^n,0) = (3*2^n-1)*4^n.
T(2^e,2) = A003951(e) for 2 <= e. Proof: T(2^e,2) = 9*8^(e-1) is a series with ratio 8 and initial term 72, as A003951(2...inf) is.
Working with consecutive powers of a prime p, we need a definition (0 <= i < e):
N(p^e,i):=#{k: 0 < k < p^e, gcd(k,p^e) = p^i} = (p-1)*p^(e-1-i). We say that these k's belong to i (respect to p^e). Note that N(p^e,0) = EulerPhi(p^e), and if 0 < k < p^e then gcd(k,p^e) = gcd(k,p^(e+1)). Let T(p^e,[i]) denote the common value of T(p^e,k)'s, where k's belong to i (q.v.PROGRAM); for example, T(p^e,[0]) = T(p^e,1). The number of the 2 X 2 matrices over Z(p^e), T(p^e,0) + Sum_{i=0..e-1} T(p^e,[i])*N(p^e,i) = p^(4e) will be useful.
On the hexagon property: Let prime p be given and let T(p^e,[0]), T(p^e,[1]), T(p^e,[2]), ..., T(p^e,[e-2]), T(p^e,[e-1]) form the e-th row of a Pascal-like triangle, e>=1. Let denote X(r,s) an element of the triangle and its value T(p^r,[s]). Let positive integers a and b given, so that the entries A(m-a,n-b), B(m-a,n), C(m,n+a), D(m+b,n+a), E(m+b,n), F(m,n-b) of the triangle form a hexagon spaced around T(p^m,[n]); if a=b=1 then they surround it. If A*C*E = B*D*F, then we say that the triangle T(.,.) has the "hexagon property". (In the case of binomial coefficients X(r,s) = COMB(r,s), the "hexagon property" holds (see [Gupta]) and moreover gcd(A,C,E) = gcd(B,D,F) (see [Hitotumatu & Sato]).)
Corollary 2.2 in [Brent & McKay] says that, for the d X d matrices over Z(p^e), (mutatis mutandis) T_d(p^e,0) = K*(1-P(d+e-1)/P(e-1)) and T_d(p^e,[i]) = K*(q^e)*((1-q^d)/(1-q))*P(d+i-1)/P(i), where q=1/p, K=(p^e)^(d^2), P(t) = Product_{j=1..t} (1-q^j), P(0):=1. (For the case d=2, we have T(p^e,[i]) = (p+1)*(p^(i+1)-1)*p^(3*e-i-2).) Due to [Brent & McKay], it can be simply proved that for d X d matrices the "hexagon property" is true. The formulation implies an obvious generalization: For the entries A(r,u), B(r,v), C(s,w), D(t,w), E(t,v), F(s,u) of the T_d(.,.)-triangle, a hexagon-like property A*C*E = B*D*F holds. This is false in general for the COMB(.,.)-triangle.
Another (rotated-hexagon-like) property: for the entries A(m-b1,n), B(m-a1,n+c2), C(m+a2,n+c2), D(m+b2,n), E(m+a2,n-c1), F(m-a1,n-c1) of the T_d(.,.)-triangle, the property A*C*E = B*D*F holds, if and only if 2*(a1 + a2) = b1 + b2. This is also in general false for COMB(.,.)-triangle.

			From _Andrew Howroyd_, Jul 16 2018: (Start)
Triangle begins:
    1;
   10,   6;
   33,  24,  24;
   88,  48,  72,  48;
  145, 120, 120, 120, 120;
  330, 144, 240, 198, 240, 144;
  385, 336, 336, 336, 336, 336, 336;
  736, 384, 576, 384, 672, 384, 576, 384;
  945, 648, 648, 864, 648, 648, 864, 648, 648;
  ... (End)

Erdos Pal, Rows n=1..100 of triangle, flattened
Richard P. Brent and Brendan D. McKay, Determinants and ranks of random matrices over Z_m, Discrete Mathematics 66 (1987) pp. 35-49.
A. K. Gupta, Generalized hidden hexagon squares, The Fibonacci Quarterly, Vol 12, Number 1, Feb.1974, pp. 45-46.
S. Hitotumatu, D. Sato, Star of David theorem (I), The Fibonacci Quarterly, Vol 13, Number 1, Feb.1975, p. 70.

Column k=0 is A020478.
Column k=1 is A000056.
Row sums are A005353.
Cf. A000252, A127917.

  (* computing T(p^e,k) ; p=prime, 1<=e, 0<=k

S(p,e)={my(u=vector(p^e)); my(t=(p-1)*p^(e-1)); u[1] = p^e + e*t; for(j=1, p^e-1, u[j+1] = t*(1+valuation(j, p))); vector(#u, k, sum(j=0, #u-1, u[j + 1]*u[(j+k-1) % #u + 1]))}
T(n)={my(f=factor(n), v=vector(n,i,1)); for(i=1, #f~, my(r=S(f[i,1], f[i,2])); for(j=0, #v-1, v[j + 1] *= r[j % #r + 1])); v}
for(n=1, 10, print(T(n))); \\ Andrew Howroyd, Jul 16 2018

T(a*b,k) = T(a,(k mod a))*T(b,(k mod b)) if gcd(a,b) = 1.

Sum_{k=1..n-1, gcd(k,n)=1} T(n,k) = A000252(n). - Andrew Howroyd, Jul 16 2018

Terms a(24)-a(55) from b-file by Andrew Howroyd, Jul 16 2018

A065430 Order of commutator subgroup of GL(2,Z_n) (invertible 2 X 2 matrices mod n: A000252).

Original entry on oeis.org

1, 3, 24, 24, 120, 72, 336, 192, 648, 360, 1320, 576, 2184, 1008, 2880, 1536, 4896, 1944, 6840, 2880, 8064, 3960, 12144, 4608, 15000, 6552, 17496, 8064, 24360, 8640, 29760, 12288, 31680, 14688, 40320, 15552, 50616, 20520, 52416, 23040, 68880

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 16 2001

This sequence may be multiplicative. - Mitch Harris, Apr 19 2005

Multiplicative because A000056 is. - Max Alekseyev

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000056, A000252, A002117.

Table[n DivisorSum[n, #^2 MoebiusMu[n/#] &]/(1 + Boole[EvenQ@ n]), {n, 41}] (* Michael De Vlieger, Mar 17 2018, after Harvey P. Dale at A000056 *)
f[p_, e_] := (p^2 - 1)*p^(3*e-2); f[2, e_] := 3*2^(3*e-3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 *)

sl(n) = n * sumdiv(n, d, d^2 * moebius(n / d));
a(n) = if (n%2, sl(n), sl(n)/2); \\ Michel Marcus, Mar 16 2018

For odd n: a(n) = A000056(n) i.e. the commutator subgroup is SL(2, Z_n);
for even n: a(n) = A000056(n) / 2 (it has index 2 in SL(2, Z_n)).
From Amiram Eldar, Nov 30 2022: (Start)
Multiplicative with a(2^e) = 3*2^(3*e-3), and a(p^e) = (p^2-1)*p^(3*e-2) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^4, where c = 11/(56*zeta(3)) = 0.1634103... . (End)

More terms from Max Alekseyev, Jan 22 2010

cp's OEIS Frontend

A316537 Number of cyclic subgroups of the group SL(2, Z(n)), counting conjugates as distinct.

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A316563 Maximum order of an element in the special linear group SL(2, Z(n)).

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A329119 Orders of the finite groups SL_2(K) when K is a finite field with q = A246655(n) elements.

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A001766 Index of (the image of) the modular group Gamma(n) in PSL_2(Z).

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A115224 Number of 3 X 3 symmetric matrices over Z(n) having determinant 1.

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A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k).

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A070732 Size of largest conjugacy class in the group GL(2,Z_n).

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