%I #30 Nov 04 2022 10:47:01
%S 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,
%T 8,30,8,20,16,24,4,36,18,24,8,40,12,42,10,16,22,46,8,36,16,32,12,52,
%U 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
%N Number of complex, weakly primitive Dirichlet characters modulo n.
%C 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".
%C Equals Mobius transform of A055653. - _Gary W. Adamson_, Feb 28 2009
%H Amiram Eldar, <a href="/A114810/b114810.txt">Table of n, a(n) for n = 1..10000</a>
%H H. Jager, <a href="http://dx.doi.org/10.1016/1385-7258(73)90069-3">On the number of Dirichlet characters with modulus not exceeding x</a>, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.
%F a(n) is multiplicative with a(p) = phi(p), a(p^k) = phi(p^k)-phi(p^(k-1)) and phi(n) = A000010(n).
%F a(n) = Sum_{d} A007431(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
%F 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
%e 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.
%t 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 *)
%t 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 *)
%Y Cf. A000010, A007431, A055231, A330523.
%Y Cf. A055653. [_Gary W. Adamson_, Feb 28 2009]
%K nonn,mult
%O 1,3
%A _Steven Finch_, Feb 19 2006