A158522 Dirichlet inverse of number of unitary divisors of n (A034444).

1, -2, -2, 2, -2, 4, -2, -2, 2, 4, -2, -4, -2, 4, 4, 2, -2, -4, -2, -4, 4, 4, -2, 4, 2, 4, -2, -4, -2, -8, -2, -2, 4, 4, 4, 4, -2, 4, 4, 4, -2, -8, -2, -4, -4, 4, -2, -4, 2, -4, 4, -4, -2, 4, 4, 4, 4, 4, -2, 8, -2, 4, -4, 2, 4, -8, -2, -4, 4, -8, -2, -4, -2, 4, -4, -4, 4, -8, -2, -4, 2, 4, -2

Offset: 1

Author

Jaroslav Krizek, Mar 20 2009

Keywords

• sign
• mult

Comments

Abs{a(n)} = A034444(n). Examples of Dirichlet convolutions with function a(n), i.e., b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A034444(n) = A063524(n), a(n) * A000005(n) = A010052(n), a(n) * A000027(n) = A074722(n), a(n) * A000012(n) = A008836(n).

Möbius transform of Liouville's lambda function (A008836). - Wesley Ivan Hurt, Jun 22 2024

Examples

			a(60) = a(2^2*3*5) = [(-1)^2*2]*[(-1)^1*2]*[(-1)^1*2] = 2*(-2)*(-2) = 8.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A034444, A063524, A000005, A010052, A000027, A074722, A001222, A001221, A000040, A006881, A120944, A000961.

Programs

Mathematica

Table[LiouvilleLambda[n] 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Mar 07 2015 *)

PARI

for(n=1,20, print1((-1)^bigomega(n)* 2^omega(n), ", ")) \\ G. C. Greubel, May 21 2017

Formula

a(n) = (-1)^A001222(n)*A034444(n) = (-1)^A001222(n)*2^A001221(n), for n >= 2.

Multiplicative with a(p^e) = 2*(-1)^e, p prime, e>0. a(p^0) = 1.

Dirichlet g.f.: zeta(2s)/(zeta(s))^2. - R. J. Mathar, Apr 02 2011

a(n) = Sum_{d|n} (-1)^Omega(d) * mu(n/d). - Wesley Ivan Hurt, Jun 22 2024