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A114810
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Number of complex, weakly primitive Dirichlet characters modulo n.
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2
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1, 1, 2, 1, 4, 2, 6, 2, 4, 4, 10, 2, 12, 6, 8, 4, 16, 4, 18, 4, 12, 10, 22, 4, 16, 12, 12, 6, 28, 8, 30, 8, 20, 16, 24, 4, 36, 18, 24, 8, 40, 12, 42, 10, 16, 22, 46, 8, 36, 16, 32, 12, 52, 12, 40, 12, 36, 28, 58, 8, 60, 30, 24, 16, 48, 20, 66, 16, 44, 24, 70, 8, 72, 36, 32, 18, 60, 24, 78
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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".
Equals Mobius transform of A055653. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 28 2009]
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REFERENCES
| 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
| a(n) is multiplicative with a(p)=phi(p), a(p^k)=phi(p^k)-phi(p^(k-1)) and phi(n)=A000010(n). a(n) = sum A007431(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
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EXAMPLE
| 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 complex Dirichlet characters in A000010.
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CROSSREFS
| Cf. A000010, A007431, A055231.
A055653 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 28 2009]
Sequence in context: A082729 A175542 A076686 * A093819 A089929 A131888
Adjacent sequences: A114807 A114808 A114809 * A114811 A114812 A114813
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KEYWORD
| nonn,mult
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AUTHOR
| S. R. Finch (Steven.Finch(AT)inria.fr), Feb 19 2006
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