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A114810 Number of complex, weakly primitive Dirichlet characters modulo n. 2
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; text; internal format)
OFFSET

1,3

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. [Gary W. Adamson, Feb 28 2009]

LINKS

Table of n, a(n) for n=1..79.

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.

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).

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.

MATHEMATICA

b[n_] := Sum[EulerPhi[d]*MoebiusMu[n/d], {d, Divisors[n]}]; squareFreeKernel[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Sum[b[n/d], {d, Divisors[Denominator[n/squareFreeKernel[n]^2]]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Sep 07 2015 *)

CROSSREFS

Cf. A000010, A007431, A055231.

Cf. A055653. [Gary W. Adamson, Feb 28 2009]

Sequence in context: A322250 A175542 A076686 * A300718 A093819 A089929

Adjacent sequences:  A114807 A114808 A114809 * A114811 A114812 A114813

KEYWORD

nonn,mult

AUTHOR

Steven Finch, Feb 19 2006

STATUS

approved

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Last modified January 17 16:11 EST 2019. Contains 319235 sequences. (Running on oeis4.)