A247257 The number of octic characters modulo n.

1, 1, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 8, 8, 8, 2, 2, 8, 4, 2, 2, 8, 4, 4, 2, 4, 4, 8, 2, 16, 4, 8, 8, 4, 4, 2, 8, 16, 8, 4, 2, 4, 8, 2, 2, 16, 2, 4, 16, 8, 4, 2, 8, 8, 4, 4, 2, 16, 4, 2, 4, 16, 16, 4, 2, 16, 4, 8, 2, 8, 8, 4, 8, 4, 4, 8, 2, 32

Offset

1,3

Comments

Number of solutions to x^8 == 1 (mod n). - Jianing Song, Nov 10 2019

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

S. Finch, Quartic and octic characters modulo n, arXiv:0907.4894 [math.NT], 2009.

Formula

Multiplicative with a(p^e) = p^min(e-1, 4) if p = 2, gcd(8, p-1) if p > 2. - Jianing Song, Nov 10 2019

Maple

A247257 := proc(n)
    local a, pf, p, r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1, pf);
        r := op(2, pf);
        if p = 2 then
            if r >= 5 then
                a := a*16 ;
            else
                a := a*op(r, [1, 2, 4, 8]) ;
            end if;
        elif modp(p, 4) = 3 then
            a := a*2;
        elif modp(p, 8) = 5 then
            a := a*4;
        elif modp(p, 8) = 1 then
            a := a*8;
        else
            error
        end if;
    end do:
    a ;
end proc:

Mathematica

g[p_, e_] := Which[p==2, 2^Min[e-1, 4], Mod[p, 4]==3, 2, Mod[p, 8]==5, 4, True, 8];
a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n];
Array[a, 80] (* Jean-François Alcover, Nov 26 2017, after Charles R Greathouse IV *)

Prog

(PARI) g(p, e)=if(p==2, 2^min(e-1, 4), if(p%4==3, 2, if(p%8==5, 4, 8)))
a(n)=my(f=factor(n)); prod(i=1, #f~, g(f[i, 1], f[i, 2])) \\ Charles R Greathouse IV, Mar 02 2015

Crossrefs

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), A319101 (k=7), this sequence (k=8).

Sequence in context: A278266 A088200 A073103 * A069177 A077659 A367953

Adjacent sequences: A247254 A247255 A247256 * A247258 A247259 A247260

Keyword

mult,nonn,easy

Author

R. J. Mathar, Mar 02 2015

Status

approved