A000252 -id:A000252 - cp's OEIS frontend

A082953 a(n) = A000252(n) / A070732(n).

1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003

From Jianing Song, Apr 20 2019: (Start)
a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)

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

Cf. A000252, A065464, A070732.
Cf. A000010, A062570, A040001.
Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
seq(A082953(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = eulerphi(n)*eulerphi(2*n); \\ Michel Marcus, Jun 04 2025

a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - Vladeta Jovovic, May 02 2005
Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011
a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 =  0.171299... . - Amiram Eldar, Oct 30 2022
a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - Ridouane Oudra, Jun 04 2025

A066514 Number of subgroups of the group GL(2,Z_n) of invertible 2 X 2 matrices mod n (sequence A000252).

Original entry on oeis.org

1, 6, 55, 234, 466, 758, 1704, 24587, 6656, 6718, 6428, 114210, 17158, 26180, 135869, 701426, 33277, 126920, 52320, 1341656, 469872, 87924, 48976, 50257306, 140918, 291060, 348528, 3952920, 175438, 3685062, 285952, 12024029, 1657948, 567994, 4624192, 15262640, 539046, 838096

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jan 04 2002

nonn

Cf. A000252.

c := CyclicGroup(n); aut := AutomorphismGroup(DirectProduct(c,c)); lat := LatticeSubgroups(aut);

[&+[s`length : s in Subgroups(GL(2,ResidueClassRing(n)))] : n in [2..10]]; // Robin Visser, Aug 09 2023

a(7)-a(11) from Andrew Howroyd, Jul 02 2018

More terms from Robin Visser, Aug 09 2023

A066541 Number of normal subgroups of the group GL(2,Z_n) of invertible 2 X 2 matrices mod n (sequence A000252).

Original entry on oeis.org

1, 3, 5, 12, 6, 17, 8, 46, 14, 22, 8, 82, 12, 28, 38, 112, 10, 48, 12, 116, 48, 28, 8, 386, 18, 44, 27, 140, 12, 162, 16, 232, 48, 38, 64, 234, 18, 42, 76, 596, 16, 196, 16, 140, 108, 28, 8, 1022, 24, 66, 66, 232, 12, 93, 64, 680, 72, 44, 8, 1032, 24, 56, 204

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jan 07 2002

nonn

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

Cf. A000252, A066514, A316498.

Concatenation([1], List([2..20], n->Size(NormalSubgroups(GL(2, Integers mod n))))); # Andrew Howroyd, Jul 04 2018

Terms a(16) and beyond from Andrew Howroyd, Jul 04 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

A065558 Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the maximal degree of an irreducible representation of G_n.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 8, 12, 12, 12, 12, 24, 14, 16, 24, 24, 18, 24, 20, 36, 32, 24, 24, 48

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 29 2001

nonn

a(n) is multiplicative and for an odd prime p : a(p) = p + 1 . The number of irreducible representations of G_n is in sequence A062354.

Conjecture: a(2n) = 2*A001615(n). - Ralf Stephan, Mar 26 2004

A000252, A062354

A068516 Number of squares (of another matrix) in the group GL(2,Z_n) described in sequence A000252.

Original entry on oeis.org

3, 16, 16, 162, 48, 696, 104, 1236, 486, 4680, 256, 9366, 2088, 2592, 1342, 28296, 3708, 44640, 2592, 11136, 14040, 97416, 1664, 100410, 28098, 99936, 11136, 250110, 7776, 327840, 20924, 74880, 84888, 112752, 19776, 671346, 133920, 149856, 16848

Offset: 2

Sharon Sela (sharonsela(AT)hotmail.com), Mar 19 2002

The sequence is multiplicative. This is the 2-dimensional analog of A046073.

Cf. A000252, A046073, A068197.

Terms a(21) and beyond from Andrey Zabolotskiy, Jul 02 2022

A069256 Size of the Sylow 2-subgroup of the group GL_2(Z_n): maximal power of 2 that divides A000252(n).

Original entry on oeis.org

1, 2, 16, 32, 32, 32, 32, 512, 16, 64, 16, 512, 32, 64, 512, 8192, 512, 32, 16, 1024, 512, 32, 32, 8192, 32, 64, 16, 1024, 32, 1024, 128, 131072, 256, 1024, 1024, 512, 32, 32, 512, 16384, 128, 1024, 16, 512, 512, 64, 64, 131072, 32, 64, 8192, 1024, 32, 32, 512

Offset: 1

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

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

Cf. A000252, A069177.

f[p_, e_] := 2^IntegerExponent[(p-1)*(p^2-1), 2]; f[2, e_] := 2^(4*e-3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1 << (4*f[i, 2]-3), 1 << valuation((f[i, 1]-1)*(f[i, 1]^2-1), 2)));} \\ Amiram Eldar, Nov 03 2023

Multiplicative with a(2^e) = 2^(4*e-3) and a(p^e) = power of 2 in prime factorization of (p - 1)*(p^2-1) for an odd prime p. - Vladeta Jovovic, Apr 17 2002

More terms from Vladeta Jovovic, Apr 17 2002

A082965 a(n) = A000252(n) / A065558(n).

Original entry on oeis.org

1, 3, 12, 16, 80, 36, 252, 128, 324, 240, 1100, 192, 1872, 756, 960, 1024, 4352, 972, 6156, 1280, 3024, 3300, 11132, 1536

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 27 2003

Cf. A000252, A065558.

A062354 a(n) = sigma(n)*phi(n).

Original entry on oeis.org

1, 3, 8, 14, 24, 24, 48, 60, 78, 72, 120, 112, 168, 144, 192, 248, 288, 234, 360, 336, 384, 360, 528, 480, 620, 504, 720, 672, 840, 576, 960, 1008, 960, 864, 1152, 1092, 1368, 1080, 1344, 1440, 1680, 1152, 1848, 1680, 1872, 1584, 2208, 1984, 2394, 1860

Offset: 1

Jason Earls, Jul 06 2001

Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the number of conjugacy classes in G_n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 13 2001
a(n) = Sum_{d|n} phi(n*d). - Vladeta Jovovic, Apr 17 2002
Apparently the Mobius transform of A062952. - R. J. Mathar, Oct 01 2011

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.

T. D. Noe, Table of n, a(n) for n=1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000
J.-L. Nicolas and Jonathan Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

Cf. A000010, A000203, A000252, A062355, A064840, A093827.

Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))

Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - Vladeta Jovovic, Apr 17 2002
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*product_{primes p} (1-p^(1-s)-p^(-s)+p^(2-2s)). - R. J. Mathar, Oct 01 2011, corrected by Vaclav Kotesovec, Dec 17 2019
6/Pi^2 < a(n)/n^2 < 1 for n > 1. - Jonathan Sondow, Mar 06 2014
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.535896... - Vaclav Kotesovec, Dec 17 2019
Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - Amiram Eldar, Aug 20 2020
a(n)/n^2 > 8/Pi^2 for odd n. - M. F. Hasler, Jul 08 2025

A053191 a(n) = n^2 * phi(n).

Original entry on oeis.org

1, 4, 18, 32, 100, 72, 294, 256, 486, 400, 1210, 576, 2028, 1176, 1800, 2048, 4624, 1944, 6498, 3200, 5292, 4840, 11638, 4608, 12500, 8112, 13122, 9408, 23548, 7200, 28830, 16384, 21780, 18496, 29400, 15552, 49284, 25992, 36504, 25600, 67240

Offset: 1

Labos Elemer, Mar 02 2000

Number of invertible 2 X 2 symmetric matrices over Z(n). - T. D. Noe, Jan 13 2006
Note that A115077 gives the number of 2 X 2 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n. - T. D. Noe, Jan 13 2006
Also Euler phi function of n^3.
For n^k, EulerPhi(n^k) = n^(k-1)*EulerPhi(n). The same holds if Phi is replaced by the cototient function.
Also, the sum of the degrees of the irreducible representations of the group GL(2,Z_n) (sequence A000252). - Sharon Sela (sharonsela(AT)hotmail.com), Feb 06 2002

			n=5: n^3 = 125, EulerPhi(125) = 125 - 25 = 100.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000252 (number of invertible 2 X 2 matrices over Z(n)), A115075, A115076, A115077.

Cf. A000010, A051953, A002618, A053650, A053192, A001248, A319087.

[ n^2*EulerPhi(n) : n in [1..100] ]; // Vincenzo Librandi, Apr 21 2011

with(numtheory):a:=n->phi(n^3): seq(a(n), n=1..41); # Zerinvary Lajos, Oct 07 2007

Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 2, 50}] (* T. D. Noe, Jan 13 2006 *)
Table[n^2*EulerPhi[n],{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

a(n) = n^2*eulerphi(n); \\ Michel Marcus, Oct 31 2017

[n^2*euler_phi(n) for n in range(1, 42)] # Zerinvary Lajos, Jun 06 2009

a(n) = n^2 * phi(n) = A000010(n^3).
Dirichlet g.f.: zeta(s-3)/zeta(s-2). - R. J. Mathar, Feb 09 2011
The n-th term of the Dirichlet inverse is n^2 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - Álvar Ibeas, Nov 24 2017
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^4 - p^3 - p + 1)) = 1.38097852211302096879... - Amiram Eldar, Dec 06 2020

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007

cp's OEIS Frontend

A082953 a(n) = A000252(n) / A070732(n).

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A066514 Number of subgroups of the group GL(2,Z_n) of invertible 2 X 2 matrices mod n (sequence A000252).

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A066541 Number of normal subgroups of the group GL(2,Z_n) of invertible 2 X 2 matrices mod n (sequence A000252).

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

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A065558 Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the maximal degree of an irreducible representation of G_n.

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A068516 Number of squares (of another matrix) in the group GL(2,Z_n) described in sequence A000252.

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A069256 Size of the Sylow 2-subgroup of the group GL_2(Z_n): maximal power of 2 that divides A000252(n).

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A082965 a(n) = A000252(n) / A065558(n).

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A062354 a(n) = sigma(n)*phi(n).

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