OFFSET
1,3
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
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).
(End)
MAPLE
MATHEMATICA
Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (* Robert G. Wilson v, Sep 06 2004 *)
PROG
(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
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Gerald McGarvey, Sep 04 2004
EXTENSIONS
More terms from Robert G. Wilson v, Sep 06 2004
Edited by N. J. A. Sloane, May 20 2006
STATUS
approved