A000056 -id:A000056 - cp's OEIS frontend

A350156 Inverse Moebius transform of A000056.

1, 7, 25, 55, 121, 175, 337, 439, 673, 847, 1321, 1375, 2185, 2359, 3025, 3511, 4897, 4711, 6841, 6655, 8425, 9247, 12145, 10975, 15121, 15295, 18169, 18535, 24361, 21175, 29761, 28087, 33025, 34279, 40777, 37015, 50617, 47887, 54625, 53119, 68881, 58975, 79465, 72655, 81433, 85015

Offset: 1

Werner Schulte, Jan 19 2022

Let f be an arbitrary arithmetic function. Define the sequence a(f; n) by a(f; n) = Sum_{i=1..n, k=1..n} f(n / gcd(gcd(i,k),n)) for n > 0. Then a(f; n) equals inverse Moebius transform of f(n) * A007434(n) for n > 0; if f is multiplicative then a(f; n) is multiplicative; this sequence uses f(n) = n (see formula section).

Column k=2 of A372968.
Cf. A000010, A000056, A000578, A001158, A007434, A055615, A057660.
Cf. A372962, A372964.

f[p_, e_] := p^(3*e) - (p - 1)*(p^(3*e) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Jan 19 2022 *)

from math import prod
from sympy import factorint
def A350156(n): return prod((q:=p**(3*e))-(p-1)*(q-1)//(p**3-1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = p^(3*e) - (p-1) * (p^(3*e) - 1) / (p^3 - 1) for prime p and e >= 0.
Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-3) * zeta(s) / zeta(s-1).
a(n) = Sum_{i=1..n, k=1..n} n / gcd(gcd(i,k),n) for n > 0.
Dirichlet convolution with A000010 equals A000578.
Dirichlet convolution of A001158 and A055615.
Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/(360*zeta(3)) = 0.225098... . - Amiram Eldar, Oct 16 2022
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_2(d^2)/sigma(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2. - Seiichi Manyama, May 25 2024

A381679 Euler transform of A000056.

Original entry on oeis.org

1, 1, 7, 31, 100, 364, 1152, 3864, 12102, 37358, 113618, 337562, 990798, 2857926, 8144334, 22902470, 63660695, 175026047, 476242001, 1283435153, 3427047146, 9072455146, 23820491998, 62057045134, 160471504373, 412022656517, 1050740365571, 2662223436203

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A061255, A381680.

Cf. A000056, A084218, A156733.

a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[4, k^2]/DivisorSigma[2, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)/sigma(k^2, 2)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^A000056(k).
G.f.: exp( Sum_{k>=1} sigma_4(k^2)/sigma_2(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2)/sigma_2(k^2) * a(n-k).
a(n) ~ exp(5*(3*zeta(5)/zeta(3))^(1/5) * n^(4/5) / 2^(7/5) - 1/10 - 12*zeta'(-3)) * A^(6/5) * (3*zeta(5)/zeta(3))^(3/25) / (2^(7/50) * sqrt(5*Pi) * n^(31/50)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 04 2025

A065501 Number of conjugacy classes in the group SL(2,Z_n) (see A000056).

Original entry on oeis.org

1, 3, 7, 10, 9, 21, 11, 30, 25, 27, 15, 70, 17, 33, 63, 76, 21, 75, 23, 90, 77, 45, 27, 210, 49, 51, 79, 110, 33, 189, 35, 168, 105, 63, 99, 250, 41, 69, 119, 270, 45, 231, 47, 150, 225, 81, 51, 532, 81, 147, 147, 170, 57, 237, 135, 330, 161, 99, 63, 630, 65, 105, 275, 352, 153

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Robin Visser, Table of n, a(n) for n = 1..500

Cf. A062354, A000056, A000252

[Nclasses(SpecialLinearGroup(2,ResidueClassRing(n))) : n in [2..50]]; // Robin Visser, Aug 06 2023

For an odd prime p : a(p) = p + 4. - Robin Visser, Aug 06 2023

a(23) corrected and more terms from Robin Visser, Aug 06 2023

A007434 Jordan function J_2(n) (a generalization of phi(n)).

Original entry on oeis.org

1, 3, 8, 12, 24, 24, 48, 48, 72, 72, 120, 96, 168, 144, 192, 192, 288, 216, 360, 288, 384, 360, 528, 384, 600, 504, 648, 576, 840, 576, 960, 768, 960, 864, 1152, 864, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 1536

Offset: 1

N. J. A. Sloane

Number of points in the bicyclic group Z/mZ X Z/mZ whose order is exactly m. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Mar 14 2006
Number of irreducible fractions among {(u+v*i)/n : 1 <= u, v <= n} with i = sqrt(-1), where a fraction (u+v*i)/n is called irreducible if and only if gcd(u, v, n) = 1. - Reinhard Zumkeller, Aug 20 2005
The weight of the n-th polynomial for the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let the weight of b1 = 1, b2 = 3, b3 = 8, b4 = 12 and let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1, and so on, be an elliptic divisibility sequence. Then weight of e2 = 4, e3 = 9, e4 = 16, e5 = 25, where weight of en is n^2 in general, while weight of bn is a(n). - Michael Somos, Aug 12 2008
J_2(n) divides J_{2k}(n). J_2(n) gives the number of 2-tuples (x1,x2), such that 1 <= x1, x2 <= n and gcd(x1, x2, n) = 1. - Enrique Pérez Herrero, Mar 05 2011
From Jianing Song, Apr 06 2019: (Start)
Let k be any quadratic field such that all prime factors of n are inert in k, O_k be the corresponding ring of integers and G(n) = (O_k/nO_k)* be the multiplicative group of integers in O_k modulo n, then a(n) is the number of elements in G(n). The exponent of G(n) is A306933(n). [Equivalently, G(p^e) can be defined as (Z_{p^2}/p^eZ_{p^2})*, where Z_{p^2} is the ring of integers of the field Q_{p^2} (with a unique maximal ideal pZ_{p^2}), and Q_{p^2} is the unique unramified quadratic extension of the p-adic field Q_p. For the group structure of G(p^e), see A306933. - Jianing Song, Jun 19 2025]
For n >= 5, a(n) is divisible by 24. (End)
The Del Centina article on page 106 mentions a formula by Halphen denoted by phi(n)T(n). - Michael Somos, Feb 05 2021

			a(4) = 12 because the divisors of 4 being 1, 2, 4, we find that phi(1)*phi(4/1)*(4/1) = 8, phi(2)*phi(4/2)*(4/2) = 2, phi(4)*phi(4/4)*(4/4) = 2 and 8 + 2 + 2 = 12.
G.f. = x + 3*x^2 + 8*x^3 + 12*x^4 + 24*x^5 + 24*x^6 + 48*x^7 + 48*x^8 + 72*x^9 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
A. Del Centina, Poncelet's porism: a long story of renewed discoveries, I, Hist. Exact Sci. (2016), v. 70, p. 106.
L. E. Dickson (1919, repr. 1971). History of the Theory of Numbers I. Chelsea. p. 147.
P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.
M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11.
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
Sukumar Das Adhikari and A. Sankaranarayanan, On an error term related to the Jordan totient function Jk(n), Journal of Number Theory Volume 34, Issue 2, February 1990, Pages 178-188.
Dorin Andrica and Mihai Piticari, On Some Extensions Of Jordan's Arithmetic Functions, Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI 2003, Alba Iulia.
E. Artal Bartolo, P. D. González Pérez, M. González Villa, and E. León-Cardenal, On a suspension formula for Denef-Loeser zeta functions, arXiv:2502.05022 [math.AG], 2025. See p. 19.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 32.
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 38.
MathOverflow, Averages of euler-phi function and similar.
Carl-Fredrik Nyberg-Brodda, On congruence subgroups of SL_2(Z[1/p]) generated by two parabolic elements, arXiv:2312.11258 [math.GR], 2023.
Nittiya Pabhapote and Vichian Laohakosol, Combinatorial Aspects of the Generalized Euler's Totient, International Journal of Mathematics and Mathematical Sciences, Volume 2010 (2010), Article ID 648165, 15 p.
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
N. J. A. Sloane, Transforms.
S. Thajoddin and S. Vangipuram, A Note On Jordan's Totient Function, Indian J. Pure Appl. Math, 1988.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Wikipedia, Jordan's totient function.
Index to divisibility sequences

Cf. A000056, A000290, A001065, A001615, A027750, A046970, A115000, A134675.
Cf. A059379 and A059380 (triangle of values of J_k(n)).
Cf. A000010 (J_1), this sequence (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A002117, A088453, A301875, A301876, A321879 (partial sums).

a007434 n = sum $ zipWith3 (\x y z -> x * y * z)
                  tdivs (reverse tdivs) (reverse divs)
                  where divs = a027750_row n;  tdivs = map a000010 divs
-- Reinhard Zumkeller, Nov 24 2012

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 2)
A007434 := proc(n)
    add(d^2*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Nov 03 2015

jordanTotient[n_, k_:1] := DivisorSum[n, #^k*MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; Table[jordanTotient[n, 2], {n, 48}] (* Enrique Pérez Herrero, Sep 14 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ n/d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], n^2 (Times @@ ((1 - 1/#[[1]]^2) & /@ FactorInteger @ n))]; (* Michael Somos, Jan 11 2014 *)
jordanTotient[n_Integer?Positive, r_:1] := DirichletConvolve[MoebiusMu[K], K^r, K, n]; Table[jordanTotient[n, 2], {n, 48}] (* Jan Mangaldan, Jun 03 2016 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 20 2004 */

{a(n) = if( n<1, 0, direuler( p=2, n, (1 - X) / (1 - X*p^2))[n])}; /* Michael Somos, Jan 11 2014 */

seq(n) = dirmul(vector(n,k,k^2), vector(n,k,moebius(k)));
seq(48)  \\ Gheorghe Coserea, May 11 2016

jordan(n,k)=my(a=n^k);fordiv(n,i,if(isprime(i),a*=(1-1/(i^k))));a  \\ Roderick MacPhee, May 05 2017

from math import prod
from sympy import factorint
def A007434(n): return prod(p**(e-1<<1)*(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

Moebius transform of squares.
Multiplicative with a(p^e) = p^(2e) - p^(2e-2). - Vladeta Jovovic, Jul 26 2001
a(n) = Sum_{d|n} d^2 * mu(n/d). - Benoit Cloitre, Apr 05 2002
a(n) = n^2 * Product_{p|n} (1-1/p^2). - Tom Edgar, Jan 07 2015
a(n) = Sum_{d|n} phi(d)*phi(n/d)*n/d; Sum_{d|n} a(d) = n^2. - Reinhard Zumkeller, Aug 20 2005
Dirichlet generating function: zeta(s-2)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Dirichlet inverse of A046970. - Michael Somos, Jan 11 2014
a(n) = a(n^2)/n^2. - Enrique Pérez Herrero, Sep 14 2010
a(n) = A000010(n) * A001615(n).
If n > 1, then 1 > a(n)/n^2 > 1/zeta(2). - Enrique Pérez Herrero, Jul 14 2011
a(n) = Sum_{d|n} phi(n^2/d)*mu(d)^2. - Enrique Pérez Herrero, Jul 24 2012
a(n) = Sum_{k = 1..n} gcd(k, n)^2 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(1) + a(2) + ... + a(n) ~ 1/(3*zeta(3))*n^3 + O(n^2). Lambert series Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x*(1 + x)/(1 - x)^3. - Peter Bala, Dec 23 2013
n * a(n) = A000056(n). - Michael Somos, Mar 20 2004
a(n) = 24 * A115000(n) unless n < 5. - Michael Somos, Aug 12 2008
a(n) = A001065(n) - A134675(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
a(n) = Sum_{k=1..n} gcd(n, k) * phi(gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 15 2018
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^2 - 1)^2) = 1.81078147612156295224312590448625180897250361794500723589001447178002894356... - Vaclav Kotesovec, Sep 19 2020
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^2 = 1/zeta(3) (A088453). - Amiram Eldar, Oct 12 2020
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*mu(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^2*mu(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} n*phi(gcd(n,k))/gcd(n,k).
a(n) = Sum_{k=1..n} phi(n*gcd(n,k))*mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} phi(n^2/gcd(n,k))*mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = Sum_{k = 1..n} phi(gcd(n, k)^2) = Sum_{d divides n} phi(d^2)*phi(n/d). - Peter Bala, Jan 17 2024
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i)*phi(j). See Tóth, p. 14. - Peter Bala, Jan 29 2024
Conjecture: a(n) = lim_{k->oo} (n^(2*(k + 1)))/A001157(n^k). - Velin Yanev, Dec 04 2024

Thanks to Michael Somos for catching an error in this sequence.

A000252 Number of invertible 2 X 2 matrices mod n.

Original entry on oeis.org

1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248, 1822176, 738720

Offset: 1

N. J. A. Sloane

For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the Product_{i=1..k} (p_i)^(e_i), then a(n) = Product_{i=1..k} (((p_i)^(2*e_i) - (p_i)^(2*e_i - 2)) * ((p_i)^(2*e_i) - (p_i)^(2*e_i - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
a(n) is the order of the automorphism group of the group C_n X C_n, where C_n is the cyclic group of order n. - Laszlo Toth, Dec 06 2011
Order of the group GL(2,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Jul 08 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.

The order of GL_2(K) for a finite field K is in sequence A059238.
Row n=2 of A316622.
Row sums of A316566.
Cf. A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A227499.

Table[n*EulerPhi[n]*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011, after Vladeta Jovovic *)

a(n)=my(f=factor(n)[,1]); n^4*prod(i=1,#f, (1-1/f[i]^2)*(1-1/f[i])) \\ Charles R Greathouse IV, Feb 06 2017

from math import prod
from sympy import factorint
def A000252(n): return prod(p**((e<<2)-3)*(p*(p*(p-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = n^4*Product_{primes p dividing n} (1 - 1/p^2)*(1 - 1/p) = n^4*Product_{primes p dividing n} p^(-3)*(p^2 - 1)*(p - 1). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
Multiplicative with a(p^e) = (p - 1)^2*(p + 1)*p^(4e-3). - David W. Wilson, Aug 01 2001
a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic, Oct 30 2001
Dirichlet g.f.: zeta(s - 4)*Product_{p prime} (1 - p^(1 - s)*(p^2 + p - 1)). - Álvar Ibeas, Nov 28 2017
a(n) = A227499(n) for odd n; (3/4)*A227499(n) for even n. - Jianing Song, Jul 08 2018
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Aug 20 2021
Sum_{n>=1} 1/a(n) = (Pi^8/3240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.2059016071... . - Amiram Eldar, Dec 03 2022

More terms from David W. Wilson, Jul 21 2001

A064767 Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

Original entry on oeis.org

1, 168, 11232, 86016, 1488000, 1886976, 33784128, 44040192, 221079456, 249984000, 2124276000, 966131712, 9726417792, 5675733504, 16713216000, 22548578304, 111203278848, 37141348608, 304812862560, 127991808000

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 24 2001

Also number of 3 X 3 invertible matrices over the ring Z/nZ. - Max Alekseyev, Nov 02 2007

Order of the group GL(3,Z_n). For n > 2, a(n) is divisible by 96. - Jianing Song, Nov 24 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29 , Iss. 1, 2005.
Index entries for sequences related to groups.

Row n=3 of A316622.
Cf. A000010, A064803.
Cf. A000252 (GL(2,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).

a[n_] := n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]]; a[1] = 1; Array[a, 20] (* Jean-François Alcover, Mar 21 2017 *)

a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k))); \\ Michel Marcus, Jun 30 2015

a(n,f=factor(n))=prod(i=1,#f~, ((1 - 1/f[i,1]^3)*(1 - 1/f[i,1]^2)*(1 - 1/f[i,1])))*n^9 \\ Charles R Greathouse IV, Mar 04 2025

from math import prod
from sympy import factorint
def A064767(n): return prod(p**(3*(3*e-2))*(p*(p*(p**2*(p*(p-1)-1)+1)+1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = phi(n)*A011785(n). - Vladeta Jovovic, Oct 29 2001
a(n) = n^9*Product_{primes p dividing n} ((1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). This also gives a formula for A011785.
Multiplicative with a(p^e) = p^(9*e-6)*(p^3 - 1)*(p^2 - 1)*(p - 1). - Vladeta Jovovic, Nov 18 2001
Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} ((p^7 - p^5 - p^4 + p^2 + p - 1)/p^7) =  0.05123382571... . - Amiram Eldar, Oct 23 2022

More terms from Vladeta Jovovic, Nov 18 2001

A305186 Number of invertible 4 X 4 matrices mod n.

Original entry on oeis.org

1, 20160, 24261120, 1321205760, 116064000000, 489104179200, 27811094169600, 86586540687360, 1044361663787520, 2339850240000000, 41393302251840000, 32053931488051200, 610296923230525440, 560671658459136000, 2815842631680000000, 5674535530486824960

Offset: 1

Jianing Song, May 27 2018

Order of the group GL(4,Z_n).
Order of the automorphism group of the group (C_n)^4, where C_n is the cyclic group of order n.
For n > 2, a(n) is divisible by 23040.

Jianing Song, Table of n, a(n) for n = 1..10000
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, arXiv:math/0605185 [math.GR], 2006.
C. J. Hillar and D. L. Rhea, Automorphisms of finite abelian groups, Amer. Math. Monthly 114 (2007), no 10, 917-923.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1, 2005.

Row n=4 of A316622.
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)).
Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)), A011786 (SL(4,Z_n)).
Cf. A000010.

{1}~Join~Array[#^16*Product[(1 - 1/p^4) (1 - 1/p^3) (1 - 1/p^2) (1 - 1/p), {p, FactorInteger[#][[All, 1]]}] &, 12, 2] (* Michael De Vlieger, May 27 2018 *)

a(n)=my(f=factor(n)[, 1]); n^16*prod(i=1, #f, (1-1/f[i]^4)*(1-1/f[i]^3)*(1-1/f[i]^2)*(1-1/f[i]))

from math import prod
from sympy import factorint
def A305186(n): return prod(p**((e<<3)-5<<1)*(p*(p*(p**3*(p**3*(p*(p-1)-1)+2)-1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = (p - 1)*(p^2 - 1)*(p^3 - 1)*(p^4 - 1)*p^(16*e-10).
a(n) = n^16*Product_{primes p dividing n} (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p).
a(n) = phi(n)*A011786(n) = A000010(n)*A011786(n).
Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} ((p^11 - p^9 - p^8 + 2*p^5 - p^2 - p + 1)/p^11) = 0.02958150406... . - Amiram Eldar, Oct 23 2022

A011785 Number of 3 X 3 matrices whose determinant is 1 mod n.

Original entry on oeis.org

1, 168, 5616, 43008, 372000, 943488, 5630688, 11010048, 36846576, 62496000, 212427600, 241532928, 810534816, 945955584, 2089152000, 2818572288, 6950204928, 6190224768, 16934047920, 15998976000, 31621943808, 35687836800

Offset: 1

Benjamin T. Love (benlove(AT)preston.polaristel.net)

Order of the group SL(3,Z_n). For n > 2, a(n) is divisible by 48. - Jianing Song, Nov 24 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Cf. A000010, A000578, A003557, A003800, A046970, A063453.
Cf. A000056 (SL(2,Z_n)), A011786 (SL(4,Z_n)).
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).

a[n_] := (n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]])/EulerPhi[n]; a[1] = 1; Array[a, 30] (* Jean-François Alcover, Mar 21 2017 *)

a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k)))/eulerphi(n); \\ Michel Marcus, Jun 30 2015

from math import prod
from sympy import factorint
def A011785(n): return prod(p**((e<<3)-5)*(p**2*(p*(p-1)*(p+1)-1)+1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Multiplicative with a(p^e) = p^(8*e-5)*(p^3 - 1)*(p^2 - 1). - Vladeta Jovovic, Nov 18 2001
For a formula see A064767.
a(n) = A046970(n)*A063453(n)*A000578(n)*A003557(n)^5. - R. J. Mathar, Mar 30 2011
a(n) = A064767(n)/phi(n). - Jianing Song, Nov 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/((p-1)^3 * (p+1)^2 * (p^2 + p + 1) * (p^6 + p^4 + p^2 + 1))) = 1.0061577672748872278355775942508642214184417621389767880397578015151659965... - Vaclav Kotesovec, Sep 19 2020
Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 - (p^3 + p^2 -1)/p^6) = 0.08630488937... . - Amiram Eldar, Oct 23 2022

More terms from John W. Layman, Feb 16 2001

Further terms from Vladeta Jovovic, Oct 29 2001

A011786 Number of 4 X 4 matrices whose determinant is 1 mod n.

Original entry on oeis.org

1, 20160, 12130560, 660602880, 29016000000, 244552089600, 4635182361600, 21646635171840, 174060277297920, 584962560000000, 4139330225184000, 8013482872012800, 50858076935877120, 93445276409856000, 351980328960000000, 709316941310853120, 2851903720876769280

Offset: 1

benlove(AT)preston.polaristel.net (Benjamin T. Love)

Order of the group SL(4,Z_n). For n > 2, a(n) is divisible by 11520. - Jianing Song, Nov 24 2018

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000056 (SL(2,Z_n)), A011785 (SL(3,Z_n)).
Cf. A000252 (GL(2,Z_n)), A064767 (GL(3,Z_n)), A305186 (GL(4,Z_n)).
Cf. A000010.

f[p_, e_] := (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2); a[1] = 1; a[n_] := n^15 * Times @@ f @@@ FactorInteger[n]; Array[a, 17] (* Amiram Eldar, Oct 23 2022 *)

a(n) = f = factor(n); n^16/eulerphi(n) * prod(i=1, #f~, (1-1/f[i,1]^4)*(1-1/f[i,1]^3)*(1-1/f[i,1]^2)*(1-1/f[i,1])); \\ Michel Marcus, Sep 02 2013

from math import prod
from sympy import factorint
def A011786(n): return prod(p**(3*(5*e-3))*(p**2*(p*(p*(p*(p*(p*(p-1)*(p+1)-1)-1)+1)+1)+1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

a(n) = (n^16/phi(n))*Product_{primes p dividing n} ((1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). Multiplicative with a(p^e) = p^(15*e-9)*(p^4 - 1)*(p^3 - 1)*(p^2 - 1). - Vladeta Jovovic, Nov 18 2001
a(n) = n^15*Product_{primes p dividing n} ((1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)) = A305186(n)/phi(n). - Jianing Song, Nov 24 2018
Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} ((p^10 - p^7 - p^6 - p^5 + p^4 + p^3 + p^2 - 1)/p^10) = 0.04715136234... . - Amiram Eldar, Oct 23 2022

More terms from Vladeta Jovovic, Nov 18 2001

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