A097945 a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010).

1, -1, -2, 0, -4, 2, -6, 0, 0, 4, -10, 0, -12, 6, 8, 0, -16, 0, -18, 0, 12, 10, -22, 0, 0, 12, 0, 0, -28, -8, -30, 0, 20, 16, 24, 0, -36, 18, 24, 0, -40, -12, -42, 0, 0, 22, -46, 0, 0, 0, 32, 0, -52, 0, 40, 0, 36, 28, -58, 0, -60, 30, 0, 0, 48, -20, -66, 0, 44, -24, -70, 0, -72, 36, 0, 0, 60, -24, -78, 0, 0, 40, -82, 0

Offset: 1

Author

Gerald McGarvey, Sep 04 2004

Keywords

• sign
• mult

Comments

Also, a(n) = mu(n)*uphi(n) where mu(n) is the Mobius function (A008683) and uphi(n) is the unitary totient function (A047994), since phi(n) = uphi(n) when n is squarefree, while mu(n) = 0 when n is not squarefree. - Franklin T. Adams-Watters, May 14 2006

Conjecture: Sum_{n>=1} mu(n)/phi(n) = Sum_{n>=1} a(n)/phi(n)^2 = 0. It is true that Sum_{n>=1} mu(n)/phi(n)^s = 0 at least for s > 1 since: phi(2)=1, phi is multiplicative, so for n's that are squarefree, the phi(n) values can be partitioned in pairs where phi(m)=phi(2m) and mu(m) = -mu(2m). So Sum_{i=1..n} mu(i)/phi(i)^s < Sum_{j=floor(n/2)..n} 1/phi(j)^s, which approaches 0 as n increases since (1) n^(1-e) < phi(n) < n for any e > 0 and n > N(e) and (2) Sum_{i..n} 1/n^s converges for s > 1. Conjecture: Sum_{n>=1} mu(n)/phi(n)^z = 0 for Re(z) > 1.

Multiplicative with a(p^1) = 1-p, a(p^e) = 0, e > 1. - Mitch Harris, May 24 2005

Row sums of triangle A143153 = a signed version of the sequence such that parity = (-) iff A008683(n) = (+); 0 or (+): (1, 1, 2, 0, 4, -2, 6, 0, 0, -4, 10, 0, 12, -6, 0, 0, 0, ...). - Gary W. Adamson, Jul 27 2008

Dirichlet inverse of A003958. - R. J. Mathar, Jul 08 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Wikipedia, Euler's totient function

Crossrefs

Cf. A000010, A003958, A007947, A008683, A008966, A023900, A047994, A143153, A173557, A322581.

Programs

Maple

with(numtheory):
a:= n-> mobius(n)*phi(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 06 2012

Mathematica

Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (* Robert G. Wilson v, Sep 06 2004 *)

PARI

a(n)=moebius(n)*eulerphi(n) \\ Charles R Greathouse IV, Feb 21 2013

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Formula

Dirichlet g.f.: Product_{primes p} (1-p^(1-s)+p^(-s)). - R. J. Mathar, Aug 29 2011

Sum_{d|n} abs(a(d)) = rad(n) = A007947(n). - Rémy Sigrist, Nov 05 2017

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = A065464/2 = (1/2) * Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.21412475283854722... Equivalently, c = A065463 * 3 / Pi^2. - Vaclav Kotesovec, Jun 14 2020

From Antti Karttunen, Aug 20 2021: (Start)

a(n) = mu(n)*A000010(n) = mu(n)*A003958(n) = mu(n)*A047994(n) = mu(n)*A173557(n), where mu is Möbius mu function (A008683).

a(n) = A008966(n) * A023900(n) = abs(mu(n)) * A023900(n).

a(n) = A322581(n) - A003958(n).

(End)

Extensions

More terms from Robert G. Wilson v, Sep 06 2004

Edited by N. J. A. Sloane, May 20 2006