A329272 Number of octic primitive Dirichlet characters modulo n.

1, 0, 1, 1, 3, 0, 1, 2, 0, 0, 1, 1, 3, 0, 3, 4, 7, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 3, 0, 1, 8, 1, 0, 3, 0, 3, 0, 3, 6, 7, 0, 1, 1, 0, 0, 1, 4, 0, 0, 7, 3, 3, 0, 3, 2, 1, 0, 1, 3, 3, 0, 0, 0, 9, 0, 1, 7, 1, 0, 1, 0, 7, 0, 0, 1, 1, 0, 1, 12, 0, 0, 1, 1, 21, 0, 3, 2, 7, 0, 3

Offset

1,5

Comments

a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a eighth-power root of unity (+-1, +-i, +-sqrt(2)/2 +- sqrt(2)/2*i, i = sqrt(-1)).

Mobius transform of A247257.

Links

Jianing Song, Table of n, a(n) for n = 1..65539

Formula

Multiplicative with a(2^e) = 2^(e-2) for 2 <= e <= 5, a(2^e) = 0 for e = 1 or e >= 6; a(p^e) = gcd(p-1, 8)-1 if p > 2 and e = 1, a(p^e) = 0 if p > 2 and e >= 2.

Example

Let w = exp(2*Pi*i/8) = sqrt(2)/2 + i*sqrt(2)/2. For n = 17, the 7 octic primitive Dirichlet characters modulo n are:
  Chi_1 = [0, 1, -i, w, -1, -w, -w^3, w^3, i, i, w^3, -w^3, -w, -1, w, -i, 1];
  Chi_2 = [0, 1, -1, i, 1, i, -i, -i, -1, -1, -i, -i, i, 1, i, -1, 1];
  Chi_3 = [0, 1, i, w^3, -1, -w^3, -w, w, -i, -i, w, -w, -w^3, -1, w^3, i, 1];
  Chi_4 = [0, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1];
  Chi_5 = [0, 1, -i, -w, -1, w, w^3, -w^3, i, i, -w^3, w^3, w, -1, -w, -i, 1];
  Chi_6 = [0, 1, -1, -i, 1, -i, i, i, -1, -1, i, i, -i, 1, -i, -1, 1];
  Chi_7 = [0, 1, i, -w^3, -1, w^3, w, -w, -i, -i, -w, w, w^3, -1, -w^3, i, 1],
so a(17) = 7.

Mathematica

f[2, e_] := If[2 <= e <= 5, 2^(e-2), 0]; f[p_, e_] := If[e == 1, GCD[p-1, 8] - 1, 0];  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

Prog

(PARI) a(n)={
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, if(e>=2&&e<=5, r*=2^(e-2), r=0; return(r)));
        if(p>2, if(e==1, r*=gcd(p-1, 8)-1, r=0; return(r)));
    );
    return(r);
}

Crossrefs

Number of k-th power primitive Dirichlet characters modulo n: A114643 (k=2), A160498 (k=3), A160499 (k=4), A307380 (k=5), A307381 (k=6), A307382 (k=7), this sequence (k=8).

Cf. A247257 (number of solutions to x^8 == 1 (mod n)).

Sequence in context: A331567 A303301 A160499 * A274876 A065718 A025428

Adjacent sequences: A329269 A329270 A329271 * A329273 A329274 A329275

Keyword

nonn,mult

Author

Jianing Song, Nov 10 2019

Status

approved