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A114810
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Number of complex, weakly primitive Dirichlet characters modulo n.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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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LINKS
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FORMULA
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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_{d} A007431(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 / 2 = 0.2679480769... . - Amiram Eldar, Nov 04 2022
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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 complex Dirichlet characters in A000010.
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MATHEMATICA
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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 *)
f[p_, e_] := If[e == 1, p - 1, (p - 1)^2*p^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2022 *)
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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