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A114643
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Number of real primitive Dirichlet characters modulo n.
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3
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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; internal format)
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OFFSET
| 1,8
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COMMENTS
| a(n)=1 if n=1; 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.
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REFERENCES
| W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, pp. 224-226.
I. J. Zucker and M. M. Robertson, Some properties of Dirichlet L-series, J. Phys. A 9 (1976) 1207-1214.
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LINKS
| Eric Weisstein's World of Mathematics, Dirichlet L-Series.
S. R. Finch, Cubic and quartic characters. [From S. R. Finch (Steven.Finch(AT)inria.fr), Jun 09 2009]
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FORMULA
| This sequence is multiplicative with a(2)=0, a(4)=1, a(8)=2, a(2^r)=0 for r>2, a(p)=1 for prime p>2 and a(p^r)=0 for r>1. - S. R. Finch (Steven.Finch(AT)inria.fr), Mar 08 2006
Dirichlet g.f. zeta(s)*(1+2^(-2s)+2^(1-3s))/ ( zeta(2s)*(1+2^(-s)) ). - R. J. Mathar, Jul 03 2011
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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)) [From S. R. Finch (Steven.Finch(AT)inria.fr), Jun 09 2009]
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CROSSREFS
| Cf. A003657, A003658.
Sequence in context: A067255 A065716 A079409 * A038498 A184154 A060952
Adjacent sequences: A114640 A114641 A114642 * A114644 A114645 A114646
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KEYWORD
| nonn,mult
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AUTHOR
| S. R. Finch (Steven.Finch(AT)inria.fr), Feb 16 2006
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