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A073103 Number of solutions to x^4 == 1 (mod n). 12
1, 1, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 8, 8, 4, 2, 2, 8, 4, 2, 2, 8, 4, 4, 2, 4, 4, 8, 2, 8, 4, 4, 8, 4, 4, 2, 8, 16, 4, 4, 2, 4, 8, 2, 2, 16, 2, 4, 8, 8, 4, 2, 8, 8, 4, 4, 2, 16, 4, 2, 4, 8, 16, 4, 2, 8, 4, 8, 2, 8, 4, 4, 8, 4, 4, 8, 2, 32, 2, 4, 2, 8, 16, 2, 8, 8, 4, 8, 8, 4, 4, 2, 8, 16, 4, 2, 4, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) = 2*A060594(n) for n = 5, 10, 13, 15, 16, 17, 20, 25, 26, 29, .... This subsequence, which contains all the primes of form 4k+1, seems to be asymptotic to 2n.

Multiplicative with a(p^e) = p^min(e-1, 3) if p = 2, 4 if p == 1 (mod 4), 2 if p == 3 (mod 4). - David W. Wilson, Jun 09 2005

Shadow transform of A123865. - Michel Marcus, Jun 06 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

S. R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2009.

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.

Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.

N. J. A. Sloane, Transforms.

FORMULA

sum(k=1, n, a(k)) seems to be asymptotic to C*n*Log(n) with C>1.4...(when sum(k=1, A060594(k)) is asymptotic to C/2*n*Log(n)).

In fact, sum(k=1, n, a(k)) is asymptotic to c*n*log(n)^2 where 2*c=0.190876.... - Steven Finch, Aug 12 2009

MAPLE

a:= n-> add(`if`(irem(j^4-1, n)=0, 1, 0), j=0..n-1):

seq(a(n), n=1..120);  # Alois P. Heinz, Jun 06 2013

# alternative

A073103 := 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

            a := a*p^min(r-1, 3) ;

        else

            if modp(p, 4) = 1 then

                a := 4*a ;

            else

                a := 2*a ;

            end if;

        end if;

    end do:

    a ;

end proc: # R. J. Mathar, Mar 02 2015

MATHEMATICA

a[n_] := Sum[If[Mod[j^4-1, n] == 0, 1, 0], {j, 0, n-1}]; Table[a[n], {n, 1, 120}] (* Jean-Fran├žois Alcover, Jun 12 2015, after Alois P. Heinz *)

PROG

(PARI) a(n)=sum(i=1, n, if((i^4-1)%n, 0, 1))

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

CROSSREFS

Cf. A060594, A060839.

Sequence in context: A327892 A278266 A088200 * A247257 A069177 A077659

Adjacent sequences:  A073100 A073101 A073102 * A073104 A073105 A073106

KEYWORD

easy,nonn,mult

AUTHOR

Benoit Cloitre, Aug 19 2002

STATUS

approved

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Last modified June 5 15:13 EDT 2020. Contains 334849 sequences. (Running on oeis4.)