login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A114643 Number of real primitive Dirichlet characters modulo n. 7
1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

a(n) = 1 if either n or -n is a fundamental discriminant (not both); a(n) = 2 if n and -n are fundamental discriminants; a(n) = 0 otherwise. Also, Sum_{k=1..n} a(k) is asymptotic to (6/Pi^2)*n.

From Jianing Song, Feb 27 2019: (Start)

If n is an odd squarefree number, then a(n) = 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(n,k)} = {Jacobi(k,n)} if n == 1 (mod 4) and {Kronecker(-n,k)} = {Jacobi(k,n)} if n == 3 (mod 4).

If n = 4*m, m is an odd squarefree number, then a(n) is also 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(-n,k)} if m == 1 (mod 4) and {Kronecker(n,k)} if m == 3 (mod 4).

If n is 8 times an odd squarefree number, then a(n) = 2, where the two real primitive Dirichlet characters modulo n are {Kronecker(n,k)} and {Kronecker(-n,k)}.

a(n) = 0 if n == 2 (mod 4), n is divisible by 16 or the square of an odd prime. (End)

Mobius transform of A060594. - Jianing Song, Mar 02 2019

REFERENCES

W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, pp. 224-226.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Steven R. Finch, Cubic and quartic characters [Broken link]

Steven R. Finch, Cubic and quartic characters

Vaclav Kotesovec, Graph - the asymptotic ratio

Eric Weisstein's World of Mathematics, Dirichlet L-Series.

I. J. Zucker and M. M. Robertson, Some properties of Dirichlet L-series, J. Phys. A 9 (1976) 1207-1214.

FORMULA

This sequence is multiplicative with a(2) = 0, a(4) = 1, a(8) = 2, a(2^r) = 0 for r > 3, a(p) = 1 for prime p > 2 and a(p^r) = 0 for r > 1. - Steven Finch, Mar 08 2006 (With correction by Jianing Song, Jun 28 2018)

Dirichlet g.f.: zeta(s)*(1 + 2^(-2s) + 2^(1-3s))/(zeta(2s)*(1 + 2^(-s))). - R. J. Mathar, Jul 03 2011

EXAMPLE

From Jianing Song, Feb 27 2019: (Start)

For n = 5, the only real primitive Dirichlet characters modulo n is {Kronecker(5,k)} = [0, 1, -1, -1, 1] = A080891, so a(5) = 1.

For n = 8, the real primitive Dirichlet characters modulo n are {Kronecker(8,k)} = [0, 1, 0, -1, 0, -1, 0, 1] = A091337 and [0, 1, 0, 1, 0, -1, 0, -1] = A188510, so a(8) = 2.

For n = 20, the only real primitive Dirichlet characters modulo n is {Kronecker(-20,k)} = [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1] = A289741, so a(20) = 1. (End)

MAPLE

A114643 := 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 =  1 then

                a := 0 ;

            elif r =  2 then

                ;

            elif r =  3 then

                a := a*2 ;

            elif r >=  4 then

                a := 0 ;

            end if;

        else

            if r =1 then

                ;

            else

                a := 0 ;

            end if;

        end if;

    end do:

    a ;

end proc:

seq(A114643(n), n=1..40) ; # R. J. Mathar, Mar 02 2015

# Alternative:

f:= proc(n) local r, v, F;

  v:= padic:-ordp(n, 2);

  if v = 1 or v >= 4 then return 0

  elif v = 3 then r:= 2

  else r:= 1

  fi;

  if numtheory:-issqrfree(n/2^v) then r else 0 fi

end proc:

map(f, [$1..100]); # Robert Israel, Oct 08 2017

MATHEMATICA

a[n_] := Sum[ MoebiusMu[n/d] * Sum[ If[ Mod[i^2 - 1, d] == 0, 1, 0], {i, 2, d}], {d, Divisors[n]}]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-Fran├žois Alcover, Jun 20 2013, after Steven Finch *)

PROG

(PARI) a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^2-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

CROSSREFS

Cf. A003657, A003658.

Cf. A160498 (number of cubic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).

Cf. A060594 (number of solutions to x^2 == 1 (mod n)).

Sequence in context: A067255 A065716 A079409 * A038498 A319510 A257217

Adjacent sequences:  A114640 A114641 A114642 * A114644 A114645 A114646

KEYWORD

nonn,mult,changed

AUTHOR

Steven Finch, Feb 16 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 03:26 EDT 2019. Contains 322380 sequences. (Running on oeis4.)