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 A114643 Number of real primitive Dirichlet characters modulo n. 8
 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-François Alcover, Jun 20 2013, after Steven Finch *) f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e >= 4, 0]; f[p_, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *) 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 AUTHOR Steven Finch, Feb 16 2006 STATUS approved

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Last modified January 27 06:18 EST 2022. Contains 350601 sequences. (Running on oeis4.)