A060594 -id:A060594 - cp's OEIS frontend

A157235 Number of primitive inequivalent oblique sublattices of hexagonal (triangular) lattice of index n (equivalence and symmetry of sublattices are determined using only parent lattice symmetries).

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 2, 1, 3, 2, 2, 2, 5, 2, 4, 3, 5, 3, 4, 4, 6, 5, 6, 4, 10, 4, 6, 6, 8, 6, 10, 5, 9, 7, 8, 6, 14, 6, 10, 10, 11, 7, 12, 8, 14, 10, 12, 8, 17, 10, 12, 11, 14, 9, 20, 9, 15, 14, 14, 12, 22, 10, 16, 14, 22, 11, 20, 11, 18, 18, 18

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 5.]

Cf. A003051 (all sublattices), A003050 (all primitive sublattices), A154272 (primitive sublattices fully inheriting the parent lattice symmetry, inlcuding the orientation of the mirrors), A000086 (primitive rotation-symmetric sublattices, counting mirror images as distinct), A060594 (primitive mirror-symmetric sublattices), A145377 (all sublattices inheriting the parent lattice symmetry), A304182.

a(n) = A003050(n) - (A000086(n)-A154272(n))/2 - A060594(n). - Andrey Zabolotskiy, Mar 19 2021

New name and a(1)=0 prepended by Andrey Zabolotskiy, May 09 2018

Terms a(31) and beyond from Andrey Zabolotskiy, Mar 19 2021

A235384 Number of involutions in the group Aff(Z/nZ).

Original entry on oeis.org

2, 4, 6, 6, 8, 8, 16, 10, 12, 12, 24, 14, 16, 24, 28, 18, 20, 20, 36, 32, 24, 24, 64, 26, 28, 28, 48, 30, 48, 32, 52, 48, 36, 48, 60, 38, 40, 56, 96, 42, 64, 44, 72, 60, 48, 48, 112, 50, 52, 72, 84, 54, 56, 72, 128, 80, 60, 60, 144, 62, 64, 80, 100, 84

Offset: 2

Tom Edgar, Jan 08 2014

Aff(Z/nZ) is the group of functions on Z/nZ of the form x->ax+b where a and b are elements of Z/nZ and gcd(a,n)=1.
Since Aff(Z/nZ) is isomorphic to the automorphism group of the dihedral group with 2n elements (when n>=3), this is the number of automorphisms of the dihedral group with 2n elements that have order 1 or 2.
The sequence is multiplicative: a(k*m) = a(k)*a(m) if m and k are coprime.
When n=26, this is the number of affine ciphers where encryption and decryption use the same function.

			Since 18 = 2*3^2, we get a(18) = 2*(3^2+1) = 20. Since 120 = 2^3*3*5, we have a(120) = (4+2^2+2^3)*(3+1)*(5+1) = 384.

Alois P. Heinz, Table of n, a(n) for n = 2..10000
K. K. A. Cunningham, Tom Edgar, A. G. Helminck, B. F. Jones, H. Oh, R. Schwell and J. F. Vasquez, On the Structure of Involutions and Symmetric Spaces of Dihedral Groups, Note di Mat., Volume 34, No. 2, 2014.

Cf. A002618, A228179, A147848, A060594, A283796, A335005.

a:= n-> add(`if`(irem(k^2, n)=1, igcd(n, k+1), 0), k=1..n-1):
seq(a(n), n=2..100);  # Alois P. Heinz, Jan 20 2014

a[n_] := Sum[If[Mod[k^2, n] == 1, GCD[n, k+1], 0], {k, 1, n-1}]; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)
f[p_, e_] := p^e + 1; f[2, 1] = 2; f[2, 2] = 6; f[2, e_] := 3*2^(e - 1) + 4; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 2]  (* Amiram Eldar, Dec 05 2022 *)

A034448(n,f=factor(n))=factorback(vector(#f~,i,f[i,1]^f[i,2]+1))
a(n)=my(m=valuation(n,2)); if(m==0,1,m==1,2,m==2,6,4+3<<(m-1))*A034448(n>>m) \\ Charles R Greathouse IV, Jul 29 2016

def a(n):
    L=list(factor(n))
    if L[0][0]==2:
        m=L[0][1]
        L.pop(0)
    else:
        m=0
    order=prod([x[0]^x[1]+1 for x in L])
    if m==1:
        order=2*order
    elif m==2:
        order=6*order
    elif m>=3:
        order=(4+2^(m-1)+2^m)*order
    return order
[a(i) for i in [2..100]]

def b(n):
    sum = 0
    for a in [x for x in range(n) if ((x^2) % n == 1)]:
        sum += gcd(a+1,n)
    return sum
[b(i) for i in [2..100]]

Suppose n = 2^m*p_1^(r_1)*p_2^(r_2)*...*p_k^(r_k) where each p_i>2 is prime, r_i>0 for all i, and m>=0 is the prime factorization of n, then:
...a(n) = Product_{1<=i<=k} (p_i^(r_i)+1) if m=0,
...a(n) = 2*Product_{1<=i<=k} (p_i^(r_i)+1) if m=1,
...a(n) = 6*Product_{1<=i<=k} (p_i^(r_i)+1) if m=2,
...a(n) = (4+2^(m-1)+2^m)*Product_{1<=i<=k} (p_i^(r_i)+1) if m>=3.
a(n) = Sum_{a in row(n) of A228179} gcd(a+1,n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(2*zeta(3)) = 0.684216... (A335005). - Amiram Eldar, Dec 05 2022

A252911 Irregular triangular array read by rows: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k, n>=1, 1<=k<=A002322(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 0, 0, 2, 1, 3, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, 1, 3, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 2, 0, 0, 2, 1, 3, 0, 4, 1, 3, 0, 4, 1, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 3, 0, 4

Offset: 1

Geoffrey Critzer, Dec 24 2014

Row sums are A000010.

Column 2 = A155828(n) = A060594(n) - 1.

			1;
1;
1, 1;
1, 1;
1, 1, 0, 2;
1, 1;
1, 1, 2, 0, 0, 2;
1, 3;
1, 1, 2, 0, 0, 2;
1, 1, 0, 2;
1, 1, 0, 0, 4, 0, 0, 0, 0, 4;
1, 3;
1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4;
1, 1, 2, 0, 0, 2;
1, 3, 0, 4;
T(15,2)=3 because the elements 4, 11, and 14 have order 2 in the modulo multiplication group (Z/15Z)*. We observe that 4^2, 11^2, and 14^2 are congruent to 1 mod 15.

Alois P. Heinz, Rows n = 1..250, flattened
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.

Cf. A000010, A002322, A054522, A060594, A155828.

with(numtheory):
T:= n-> `if`(n=1, 1, (p-> seq(coeff(p, x, j), j=1..degree(p)))(
         add(`if`(igcd(n, i)>1, 0, x^order(i, n)), i=1..n-1))):
seq(T(n), n=1..30);  # Alois P. Heinz, Dec 30 2014

Table[Table[
   Count[Table[
     MultiplicativeOrder[a, n], {a,
      Select[Range[n], GCD[#, n] == 1 &]}], k], {k, 1,
    CarmichaelLambda[n]}], {n, 1, 20}] // Grid

A327925 Irregular table read by rows: T(m,n) is the number of non-isomorphic groups G such that G is the semidirect product of C_m and C_n, where C_m is a normal subgroup of G and C_n is a subgroup of G, 1 <= n <= A002322(m).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 4, 1, 6, 1, 4, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 1, 6

Offset: 1

Jianing Song, Sep 30 2019

The semidirect product of C_m and C_n has group representation G = , where r is any number such that r^n == 1 (mod m). Two groups G =  and G' =  are isomorphic if and only if there exists some k, gcd(k,n) = 1 such that r^k == s (mod m), in which case f(x^i*y^j) = x^i*y^(k*j) is an isomorphic mapping from G to G'.
Given m, T(m,n) only depends on the value of gcd(n,psi(m)), psi = A002322 (Carmichael lambda). So each row of A327924 is periodic with period psi(m), so we have this for an alternative version.
Every number k occurs in the table. By Dirichlet's theorem on arithmetic progressions, there exists a prime p such that p == 1 (mod 2^(k-1)), then T(p,2^(k-1)) = d(gcd(2^(k-1),p-1)) = k (see the formula below). For example, T(5,4) = 3, T(17,8) = 4, T(17,16) = 5, T(97,32) = 6, T(193,64) = 7, ...
Row m and Row m' are the same if and only if (Z/mZ)* = (Z/m'Z)*, where (Z/mZ)* is the multiplicative group of integers modulo m. The if part is clear; for the only if part, note that the two sequences {(number of x in (Z/mZ)* such that x^n = 1)}{n>=1} and {T(m,n)}{n>=1} determine each other, and the structure of a finite abelian group G is uniquely determined by the sequence {(number of x in G such that x^n = 1)}{n>=1}. - _Jianing Song, May 16 2022

			Table starts
m = 1: 1;
m = 2: 1;
m = 3: 1, 2;
m = 4: 1, 2;
m = 5: 1, 2, 1, 3;
m = 6: 1, 2;
m = 7: 1, 2, 2, 2, 1, 4;
m = 8: 1, 4;
m = 9: 1, 2, 2, 2, 1, 4;
m = 10: 1, 2, 1, 3;
m = 11: 1, 2, 1, 2, 2, 2, 1, 2, 1, 4;
m = 12: 1, 4;
m = 13: 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6;
m = 14: 1, 2, 2, 2, 1, 4;
m = 15: 1, 4, 1, 6;
m = 16: 1, 4, 1, 6;
m = 17: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5;
m = 18: 1, 2, 2, 2, 1, 4;
m = 19: 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 1, 6;
m = 20: 1, 4, 1, 6;
Example shows that T(21,6) = 6: The semidirect product of C_21 and C_6 has group representation G = <x, y|x^21 = y^6 = 1, yxy^(-1) = x^r>, where r = 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20. Since 2^5 == 11 (mod 21), 4^5 == 16 (mod 21), 5^5 == 17 (mod 21), 10^5 == 19 (mod 21), there are actually four pairs of isomorphic groups, giving a total of 8 non-isomorphic groups.

Jianing Song, Table of n, a(n) for n = 1..8346 (the first 200 rows)
Math Overflow, When are two semidirect products of two cyclic groups isomorphic

Cf. A327925, A002322, A000010, A000005.

Cf. also A060594, A060839, A073103, A319099, A319100, A319101, A051194.

numord(n,q) = my(v=divisors(q),r=znstar(n)[2]); sum(i=1,#v,prod(j=1,#r,gcd(v[i],r[j]))*moebius(q/v[i]))
T(m,n) = my(u=divisors(n)); sum(i=1,#u,numord(m,u[i])/eulerphi(u[i]))
Row(m) = my(l=if(m>2,znstar(m)[2][1],1), R=vector(l,n,T(m,n))); R

T(m,n) = Sum_{d|n} (number of elements x such that ord(x,m) = d)/phi(d), where ord(x,m) is the multiplicative order of x modulo m, phi = A000010.
Equivalently, T(m,n) = Sum_{d|gcd(n,psi(m))} (number of elements x such that ord(x,m) = d)/phi(d). - Jianing Song, May 16 2022
For odd primes p, T(p^e,n) = d(gcd(n,(p-1)*p^(e-1))) = A051194((p-1)*p^(e-1),n), d = A000005; for e >= 3, T(2^e,n) = 2*(v2(n)+1) for even n and 1 for odd n, where v2 is the 2-adic valuation.

A343292 Number of distinct results produced when generating a graphical image of each row of the multiplication table modulo n.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 7, 9, 7, 12, 8, 12, 13, 14, 10, 16, 11, 18, 17, 18, 13, 24, 16, 21, 19, 24, 16, 28, 17, 26, 25, 27, 25, 32, 20, 30, 29, 36, 22, 38, 23, 36, 35, 36, 25, 44, 29, 41, 37, 42, 28, 46, 37, 48, 41, 45, 31, 56, 32, 48, 47, 50, 43, 58, 35, 54, 49, 60

Offset: 1

Sébastien Dumortier, Apr 12 2021

nonn

The k-th row of the multiplication tables can be shown graphically by drawing a line for each i from i to k * i (mod n). The direction of the lines is not important.

			Modulo 11, the 2 and 6 time tables, the 3 and 4 time tables, the 5 and 9 time tables, and the 7 and 8 time tables give the same pattern. So there are only 7 different time tables (0,1,2,3,5,7 and 10).

Michael De Vlieger, Scatterplot of (n, a(n)) for n=1..2^16.
Michael De Vlieger, Annotated scatterplot of (n, a(n)) for n=1..240, labeling a(n), with color function related to ratio (a(n)+(n+3)/2)/((n-3)/2), black for prime n. Red dashed line has slope 1. Blue dashed line = (a(n)+3)/2.
Michael De Vlieger, Annotated scatterplot of (n, (a(n)+(n+3)/2)/((n-3)/2)) for n=1..2^12, the highest values are labeled a(n).
Steve Phelps, Modular Times Table, GeoGebra.

Cf. A000010, A018253, A060594, A329152.

{1}~Join~Array[# - (EulerPhi[#] - Sum[Boole[Mod[k^2, #] == 1], {k, #}])/2 &, 69, 2] (* Michael De Vlieger, Apr 13 2021 *)

G(n,r)={Set(vector(n, i, my(j=i*r%n); [min(i,j), max(i,j)]))}
a(n)={#Set(vector(n, k, concat(G(n,k-1))))} \\ Andrew Howroyd, Apr 12 2021

\\ here b(n) is A060594(n).
b(n)={my(o=valuation(n, 2)); 2^(omega(n>>o)+max(min(o-1, 2), 0))}
a(n)={n - (eulerphi(n)-b(n))/2} \\ Andrew Howroyd, Apr 12 2021

a(n) = n - A329152(n) = n - (A000010(n) - A060594(n))/2. - Andrew Howroyd, Apr 12 2021

a(p) = (p + 3)/2 for p prime. - Michael De Vlieger, Apr 13 2021

A358016 a(n) is the largest k <= n-2 such that k^2 == 1 (mod n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 11, 9, 1, 1, 1, 11, 13, 1, 1, 19, 1, 1, 1, 15, 1, 19, 1, 17, 23, 1, 29, 19, 1, 1, 25, 31, 1, 29, 1, 23, 26, 1, 1, 41, 1, 1, 35, 27, 1, 1, 34, 43, 37, 1, 1, 49, 1, 1, 55, 33, 51, 43, 1, 35, 47, 41, 1, 55, 1, 1, 49, 39, 43, 53

Offset: 3

Darío Clavijo, Oct 24 2022

nonn

It appears that the terms > 1 comprise the sequence A277777.

What is the asymptotic behavior of this sequence? Numerically, it seems like sum_{3 <= n <= x} a(n) is of the order x log x or so. - Charles R Greathouse IV, Oct 27 2022

Charles R Greathouse IV, Table of n, a(n) for n = 3..10000
Project Euler, Problem 451. Modular inverses, where a(n) = I(n).

Cf. A228179, A277777, A033948, A060594.

lkn[n_]:=Module[{k=n-2},While[PowerMod[k,2,n]!=1,k--];k]; Array[lkn,80,3] (* Harvey P. Dale, Sep 01 2023 *)

a(n) = forstep(k=n-2, 1, -1, if ((gcd(k, n) == 1) && (lift(Mod(1,n)/k) == k), return(k));); \\ Michel Marcus, Oct 25 2022

rootsOfUnity(p,e,pe=p^e)=if(p>2, return(Mod([1,-1],pe))); if(e==1, return(Mod([1],2))); if(e==2, return(Mod([1,3],4))); Mod([1,pe/2-1,pe/2+1,-1],pe)
a(n,f=factor(n))=my(v=apply(x->rootsOfUnity(x[1],x[2]),Col(f)),r,t); forvec(u=vector(#v,i,[1,#v[i]]), t=lift(chinese(vector(#u,i,v[i][u[i]]))); if(t>r && tCharles R Greathouse IV, Oct 26 2022

def a(n):
  for k in range(n - 2, 0, -1):
    if pow(k,2,n) == 1: return k

from sympy.ntheory import sqrt_mod_iter
def A358016(n): return max(filter(lambda k: k<=n-2,sqrt_mod_iter(1,n))) # Chai Wah Wu, Oct 26 2022

A387317 Number of good involutions of all nontrivial linear quandles of order n.

Original entry on oeis.org

1, 4, 1, 2, 1, 44, 1, 2, 1, 414, 1, 2, 31, 5784, 1, 2, 1, 97358, 237, 2, 1, 1917064, 1, 2, 1, 42406158, 1

Offset: 3

Luc Ta, Aug 26 2025

A linear quandle is a pair (Z/nZ, k) where k is a unit in Z/nZ, viewed as an Alexander quandle under the operation a(b) := ka + (1-k)b. A linear quandle is trivial if and only if k = 1.

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). The pair (Q,f) is called a symmetric quandle.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Lực Ta, Good involutions of twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025. See Table 2.
Lực Ta, Symmetric-Linear-Quandles, GitHub, 2025.
Index entries for sequences related to quandles and racks

Cf. A060594, A202828, A386233, A386234.

GAP
```
See Ta, GitHub link
```

If A060594(n) = 2, then a(n) = 1 if n is odd, a(n) = 4 if n = 4, and a(n) = 2 otherwise. See Ta, Ex. 5.8 and Prop. 5.9.

For all n >= 1, we have a(4n) >= A202828(n), with equality if and only if n = 1. See Ta, Thm. 5.11.

Some terms corrected by Luc Ta, Sep 03 2025

A114811 Number of real, weakly primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 4, 0, 2, 0, 2, 2, 4, 2, 2, 4, 0, 2, 0, 2, 2, 4, 2, 0, 4, 2, 4, 0, 2, 2, 4, 4, 2, 4, 2, 2, 0, 2, 2, 0, 0, 0, 4, 2, 2, 0, 4, 4, 4, 2, 2, 4, 2, 2, 0, 0, 4, 4, 2, 2, 4, 4, 2, 0, 2, 2, 0, 2, 4, 4, 2, 0, 0, 2, 2, 4, 4, 2, 4, 4, 2, 0, 4, 2, 4, 2, 4, 0, 2, 0, 0, 0, 2, 4, 2, 4, 8

Offset: 1

Steven Finch, Feb 19 2006

Any primitive Dirichlet character is weakly primitive (not conversely). Jager uses the phrase "proper character", but this conflicts with other authors (e.g., W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, p. 224) who use the word "proper" to mean the same as "primitive".

			 The function chi defined on the integers by chi(1)=1, chi(5)=-1 and chi(2)=chi(3)=chi(4)=chi(6)=0 [and extended periodically] is a weakly primitive character mod 6, but not mod 12 or mod 18. In this sense, we eliminate the "overcounting" of real Dirichlet characters in A060594.

Antti Karttunen, Table of n, a(n) for n = 1..65537
H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76 = Indag. Math. 35 (1973) 452-455.

Cf. A055231, A060594, A114643.

rad[n_] := Times @@ First /@ FactorInteger[n]; A055231[n_] := Denominator[ n/rad[n]^2 ]; A114643[n_] := Sum[ MoebiusMu[n/d] * Sum[ If[ Mod[i^2 - 1, d] == 0, 1, 0], {i, 2, d}], {d, Divisors[n] } ]; A114643[1] = 1; a[n_] := Sum[ A114643[n/d], {d, Divisors[ A055231[n] ] } ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 20 2013 *)
f[p_, e_] := 0; f[p_, 1] = 2; f[2, e_] := 0; f[2, 1] = f[2, 2] = 1; f[2, 3] = 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 11 2023 *)

A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231
A114643(n) = sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^2-1)%d, 0, 1)), 0)); \\ From A114643
A114811(n) = sumdiv(A055231(n),d,A114643(n/d)); \\ Antti Karttunen, Sep 27 2018

A114811(n) = { my(f=factor(n)); for(i=1,#f~,if(2==f[i,1],if(f[i,2]<3,f[i,2]=0,if(3==f[i,2],f[i,2]=1,f[i,1]=0)),if(f[i,2]>1,f[i,1]=0,f[i,1]=2))); factorback(f); }; \\ Antti Karttunen, Sep 27 2018, after Steven Finch's multiplicative formula.

a(n) = sum A114643(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).

This sequence is multiplicative with a(2)=1, a(4)=1, a(8)=2, a(2^r)=0 for r>2, a(p)=2 for prime p>2 and a(p^r)=0 for r>1. - Steven Finch, Mar 08 2006

A185278 Number of isomorphism classes of generalized Petersen graphs G(n,k) on 2n vertices with gcd(n,k) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 2, 3, 3, 5, 2, 5, 3, 4, 3, 6, 4, 6, 4, 5, 4, 8, 3, 8, 5, 6, 5, 7, 4, 10, 5, 7, 6, 11, 4, 11, 6, 7, 6, 12, 6, 11, 6, 9, 7, 14, 5, 11, 8, 10, 8, 15, 6, 16, 8, 10, 9, 14, 6, 17, 9, 12, 7, 18, 8, 19, 10, 11, 10, 16, 7, 20, 10, 14, 11, 21, 8, 18, 11, 15, 12, 23, 7, 19, 12, 16, 12, 19, 10, 25, 11, 16, 11

Offset: 3

N. J. A. Sloane, Feb 19 2011

Robert Israel, Table of n, a(n) for n = 3..10000
Marko Petkovsek and Helena Zakrajsek, Enumeration of I-graphs: Burnside does it again, Ars Mathematica Contemporanea, 2 (2009) 241-262.
A. Steimle and W. Staton, The isomorphism classes of the generalized Petersen graphs, Discrete Math. 309 (2009), 231-237.

Cf. A000010, A000089, A060594.

# using functions A060594 and A000089 as defined in those sequences
f:= n -> (numtheory:-phi(n)+A060594(n)+A000089(n))/4:
map(f, [$3..100]); # Robert Israel, Sep 06 2018

a(n) = (A000010(n) + A060594(n) + A000089(n))/4.

A300915 Order of the group PSL(2,Z_n).

Original entry on oeis.org

1, 6, 12, 24, 60, 72, 168, 96, 324, 360, 660, 288, 1092, 1008, 720, 768, 2448, 1944, 3420, 1440, 2016, 3960, 6072, 1152, 7500, 6552, 8748, 4032, 12180, 4320, 14880, 6144, 7920, 14688, 10080, 7776, 25308, 20520, 13104, 5760

Offset: 1

Geoffrey Critzer, Mar 16 2018

The projective special linear group PSL(2,Z_n) is the quotient group of SL(2,Z_n) with its center. The center of SL(2,Z_n) is the group of scalar matrices whose diagonal entry is x in Z_n such that x^2 = 1. The elements of PSL(2,Z_n) are equivalence classes of 2 X 2 matrices with entries in Z_n where two matrices are equivalent if one is a scalar multiple of the other.

Andrew Howroyd, 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. A000056, A060594.

n := 2; nn = 40; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; Prepend[ Table[Product[ FactorInteger[m][[All, 1]][[j]]^(n^2 (FactorInteger[m][[All, 2]][[j]] - 1)) \[Gamma][n,FactorInteger[m][[All, 1]][[j]]], {j, 1, PrimeNu[m]}], {m, 2, nn}]/Table[EulerPhi[m], {m, 2, nn}]/ Table[Count[Mod[Select[Range[m], GCD[#, m] == 1 &]^n, m], 1], {m, 2, nn}], 1]
f[p_, e_] := (p^2-1)*p^(3*e-2)/2; f[2, e_] := Switch[e, 1, 6, 2, 24, , 3*2^(3*e-4)]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Dec 01 2022 *)

a(n) = {my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); (p^2-1)*p^(3*e-2)/if(p==2, 2^min(2, e-1), 2))} \\ Andrew Howroyd, Aug 01 2018

a(n) = A000056(n)/A060594(n).

Multiplicative with a(2) = 6, a(2^2) = 24, a(2^e) = 3*2^(3*e-4) for e > 2, and a(p^e) = (p^2-1)*p^(3*e-2)/2 for p > 2. - Amiram Eldar, Dec 01 2022

Keyword:mult added by Andrew Howroyd, Aug 01 2018

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