A276833 Sum of mu(d)*phi(d) over divisors d of n.

1, 0, -1, 0, -3, 0, -5, 0, -1, 0, -9, 0, -11, 0, 3, 0, -15, 0, -17, 0, 5, 0, -21, 0, -3, 0, -1, 0, -27, 0, -29, 0, 9, 0, 15, 0, -35, 0, 11, 0, -39, 0, -41, 0, 3, 0, -45, 0, -5, 0, 15, 0, -51, 0, 27, 0, 17, 0, -57, 0, -59, 0, 5, 0, 33, 0, -65, 0, 21, 0, -69, 0, -71, 0, 3, 0, 45, 0, -77, 0, -1, 0, -81, 0, 45, 0, 27, 0, -87, 0, 55, 0, 29, 0, 51, 0, -95, 0, 9

Offset

1,5

Comments

Discovered when incorrectly applying Mobius inversion formula.

a(n)*a(m) = a(n*m) if gcd(n,m)=1 (has a simple proof).

Strongly multiplicative: a(p^e) = 2 - p. - Charles R Greathouse IV, Oct 01 2019

Links

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Formula

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

G.f.: Sum_{k>=1} mu(k)*phi(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018

a(n) = Product_{p prime and p|n} (2-p). - Robert FERREOL, Mar 14 2020

Dirichlet g.f.: zeta(s) * Product_{primes p} (1 - p^(1-s) + p^(-s)). - Vaclav Kotesovec, Jun 14 2020

a(n) = Sum_{k = 1..n} mu(lcm(k, n)/k). - Peter Bala, Jan 16 2024

Example

mu(d)*phi(d) = 1*1,-1*1,-1*2, 1*2 for d=1,2,3,6, so a(6) = 1*1-1*1-1*2+1*2 = 0.

Maple

with(numtheory):seq(convert(map(x->2-x, factorset(n)), `*`), n=1..99); # Robert FERREOL, Mar 14 2020

Mathematica

Table[Sum[MoebiusMu[d] EulerPhi[d], {d, Divisors[n]}], {n, 99}] (* Indranil Ghosh, Mar 10 2017 *)
a[1] = 1; a[n_] := Times @@ ((2 - First[#])& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

Prog

(PARI) r=0; fordiv(n, d, r+=moebius(d)*eulerphi(d)); r
(PARI) a(n) = sumdiv(n, d, moebius(d)*eulerphi(d)); \\ Michel Marcus, Sep 30 2016
(PARI) a(n)=my(f=factor(n)[, 1]); prod(i=1, #f, 2-f[i]) \\ Charles R Greathouse IV, Oct 01 2019
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Crossrefs

For squarefree numbers, the absolute value is equal to A166586 (first exception at 25).

Cf. A097945.

Sequence in context: A234434 A234020 A348259 * A166586 A122274 A373023

Adjacent sequences: A276830 A276831 A276832 * A276834 A276835 A276836

Keyword

mult,sign,easy

Author

Jurjen N.E. Bos, Sep 20 2016

Status

approved