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Search: id:A114811
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| A114811 |
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Number of real, weakly primitive Dirichlet characters modulo n. |
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+0
1
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| 1, 1, 2, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 4, 0, 2, 0, 2, 2, 4, 2, 2, 4, 0, 2, 0, 2, 2, 4, 2, 0, 4, 2, 4, 0, 2, 2, 4, 4, 2, 4, 2, 2, 0, 2, 2, 0, 0, 0, 4, 2, 2, 0, 4, 4, 4, 2, 2, 4, 2, 2, 0, 0, 4, 4, 2, 2, 4, 4, 2, 0, 2, 2, 0, 2, 4, 4, 2, 0, 0, 2, 2, 4, 4, 2, 4, 4, 2, 0, 4, 2, 4, 2, 4, 0, 2, 0, 0, 0, 2, 4, 2, 4, 8
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OFFSET
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1,3
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COMMENT
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Any primitive Dirichlet character is weakly primitive (not conversely). Jager uses the phrase "proper character", but this conflicts with other authors (e.g., W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, p. 224) who use the word "proper" to mean the same as "primitive".
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REFERENCES
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H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.
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FORMULA
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a(n) = sum A114643(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
This sequence is multiplicative with a(2)=1, a(4)=1, a(8)=2, a(2^r)=0 for r>2, a(p)=2 for prime p>2 and a(p^r)=0 for r>1. - S. R. Finch (Steven.Finch(AT)inria.fr), Mar 08 2006
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EXAMPLE
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The function chi defined on the integers by chi(1)=1, chi(5)=-1 and
chi(2)=chi(3)=chi(4)=chi(6)=0 [and extended periodically] is a
weakly primitive character mod 6, but not mod 12 or mod 18. In this
sense, we eliminate the "overcounting" of real Dirichlet characters
in A060594.
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CROSSREFS
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Cf. A055231, A060594, A114643.
Sequence in context: A098965 A016443 A120256 this_sequence A043531 A043556 A057226
Adjacent sequences: A114808 A114809 A114810 this_sequence A114812 A114813 A114814
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KEYWORD
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nonn,mult
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AUTHOR
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S. R. Finch (Steven.Finch(AT)inria.fr), Feb 19 2006
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