This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A114810 #30 Nov 04 2022 10:47:01
%S A114810 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 A114810 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 A114810 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 A114810 Number of complex, weakly primitive Dirichlet characters modulo n.
%C A114810 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 A114810 Equals Mobius transform of A055653. - _Gary W. Adamson_, Feb 28 2009
%H A114810 Amiram Eldar, <a href="/A114810/b114810.txt">Table of n, a(n) for n = 1..10000</a>
%H A114810 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 A114810 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 A114810 a(n) = Sum_{d} A007431(n/d), where the sum is over all divisors 1<=d<=n of A055231(n).
%F A114810 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 A114810 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 A114810 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 A114810 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 A114810 Cf. A000010, A007431, A055231, A330523.
%Y A114810 Cf. A055653. [_Gary W. Adamson_, Feb 28 2009]
%K A114810 nonn,mult
%O A114810 1,3
%A A114810 _Steven Finch_, Feb 19 2006