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

1,3

Comments

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

Links

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000010, A002618, A057660, A007434.

Sequence in context: A056519 A295972 A333405 * A257331 A240400 A031448

Adjacent sequences: A063442 A063443 A063444 * A063446 A063447 A063448

Keyword

nonn,mult,easy

Author

Labos Elemer, Jul 24 2001

Status

approved