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

Author

R. J. Mathar, Mar 02 2015

Keywords

• mult
• nonn
• easy

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

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).

Programs

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 *)

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

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