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

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

2,1

Comments

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.

Links

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.

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(Sage)
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]]
(Sage)
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]]
(PARI) 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

Crossrefs

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

Sequence in context: A267460 A092989 A065558 * A342597 A035280 A244367

Adjacent sequences: A235381 A235382 A235383 * A235385 A235386 A235387

Keyword

nonn,easy,look,mult,nice

Author

Tom Edgar, Jan 08 2014

Status

approved