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
Amiram Eldar, Table of n, a(n) for n = 1..10000
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).
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
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 *)
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 *)
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Steven Finch, Feb 19 2006
STATUS
approved