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
KEYWORD
nonn,mult
AUTHOR
Jianing Song, Nov 10 2019
STATUS
approved