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

1, 1, 5, 6, 19, 5, 41, 24, 48, 19, 109, 30, 155, 41, 95, 96, 271, 48, 341, 114, 205, 109, 505, 120, 480, 155, 432, 246, 811, 95, 929, 384, 545, 271, 779, 288, 1331, 341, 775, 456, 1639, 205, 1805, 654, 912, 505, 2161, 480, 2016, 480, 1355, 930, 2755, 432

Offset: 1

Author

Labos Elemer, Jul 24 2001

Keywords

• nonn
• mult
• easy

Comments

Same as Moebius transform of g(x) = x*EulerPhi(x). - Benoit Cloitre, Apr 05 2002

Examples

			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.
a(20) = a(4)*a(5) = (16 - 8 - 4 + 2)*(25 - 5 - 1) = 6*19 = 114.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A002618, A057660, A007434.

Programs

Mathematica

Table[Sum[EulerPhi[d]*MoebiusMu[n/d]*d, {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Feb 01 2019 *)

PARI

a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)*moebius(n/d)))

Formula

a(n) = Sum_{d|n} phi(d^2)*mu(n/d).

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

Dirichlet g.f. zeta(s-2)/(zeta(s)*zeta(s-1)). - R. J. Mathar, Feb 09 2011

Sum_{k=1..n} a(k) ~ 2*n^3 / (Pi^2 * Zeta(3)). - Vaclav Kotesovec, Feb 01 2019

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

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