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A063445 Moebius transform of f(x) = EulerPhi(x^2) function (A002618).

%I A063445 #33 Jan 22 2024 00:04:59
%S A063445 1,1,5,6,19,5,41,24,48,19,109,30,155,41,95,96,271,48,341,114,205,109,
%T A063445 505,120,480,155,432,246,811,95,929,384,545,271,779,288,1331,341,775,
%U A063445 456,1639,205,1805,654,912,505,2161,480,2016,480,1355,930,2755,432
%N A063445 Moebius transform of f(x) = EulerPhi(x^2) function (A002618).
%C A063445 Same as Moebius transform of g(x) = x*EulerPhi(x). - _Benoit Cloitre_, Apr 05 2002
%H A063445 Amiram Eldar, <a href="/A063445/b063445.txt">Table of n, a(n) for n = 1..10000</a>
%F A063445 a(n) = Sum_{d|n} phi(d^2)*mu(n/d).
%F A063445 Multiplicative with a(p) = p^2 - p - 1 and a(p^e) = p^(2*e) - p^(2*e-1) - p^(2*e-2) + p^(2*e-3), e > 1. - _Vladeta Jovovic_, Jul 29 2001
%F A063445 Dirichlet g.f. zeta(s-2)/(zeta(s)*zeta(s-1)). - _R. J. Mathar_, Feb 09 2011
%F A063445 Sum_{k=1..n} a(k) ~ 2*n^3 / (Pi^2 * Zeta(3)). - _Vaclav Kotesovec_, Feb 01 2019
%F A063445 Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2-p-1) + p/((p-1)^3 * (p+1)^2)) = 3.037448431566721466562170968413075105160439538735056586164601312913619316... - _Vaclav Kotesovec_, Sep 20 2020
%F A063445 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)*moebius(gcd(i, j, n)) = Sum_{d divides n} d*moebius(d)*J_2(n/d), where J_2 is the Jordan totient function A007434. - _Peter Bala_, Jan 21 2024
%e A063445 For n=20, divisors = {1,2,4,5,10,20}, phi(d^2) = {1,2,8,20,40,160}, mu(20/d) = {0,1,-1,0,-1,1}, a(20) = 0 + 2 - 8 + 0 - 40 + 160 = 114.
%e A063445 a(20) = a(4)*a(5) = (16 - 8 - 4 + 2)*(25 - 5 - 1) = 6*19 = 114.
%t A063445 Table[Sum[EulerPhi[d]*MoebiusMu[n/d]*d, {d, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, Feb 01 2019 *)
%o A063445 (PARI) a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)*moebius(n/d)))
%Y A063445 Cf. A000010, A002618, A057660, A007434.
%K A063445 nonn,mult,easy
%O A063445 1,3
%A A063445 _Labos Elemer_, Jul 24 2001