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A247257 The number of octic characters modulo n. 7
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A212595

Adjacent sequences:  A247254 A247255 A247256 * A247258 A247259 A247260

KEYWORD

mult,nonn,easy

AUTHOR

R. J. Mathar, Mar 02 2015

STATUS

approved

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Last modified September 25 20:01 EDT 2020. Contains 337344 sequences. (Running on oeis4.)