A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

1, -2, -2, 0, -2, 4, -2, 0, 0, 4, -2, 0, -2, 4, 4, 0, -2, 0, -2, 0, 4, 4, -2, 0, 0, 4, 0, 0, -2, -8, -2, 0, 4, 4, 4, 0, -2, 4, 4, 0, -2, -8, -2, 0, 0, 4, -2, 0, 0, 0, 4, 0, -2, 0, 4, 0, 4, 4, -2, 0, -2, 4, 0, 0, 4, -8, -2, 0, 4, -8, -2, 0, -2, 4, 0, 0, 4, -8, -2, 0, 0, 4, -2, 0, 4, 4, 4, 0, -2, 0, 4, 0, 4, 4, 4, 0, -2, 0, 0, 0, -2, -8, -2, 0, -8

Offset: 1

Author

Wesley Ivan Hurt, May 29 2013

Keywords

• sign
• mult

Comments

The prime numbers are the only solutions to mu(n)*d(n) = -2.

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

The Moebius inverse is A076479, and the Dirichlet inverse A061142. - R. J. Mathar, Jun 03 2013

Möbius transform of (-1)^omega(n). - Wesley Ivan Hurt, Jun 22 2024

Examples

			a(5) = mu(5)*d(5) = (-1)(2) = -2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000005, A000040, A001358, A008683, A074823 (absolute values), A001221.

Programs

Maple

with(numtheory); a:=n->mobius(n)*tau(n); seq(a(k),k=1..100);

Mathematica

Table[MoebiusMu[n] DivisorSigma[0, n], {n, 105}] (* Michael De Vlieger, Jul 23 2017 *)

PARI

A226177(n) = moebius(n)*numdiv(n); \\ Antti Karttunen, Jul 23 2017

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X))[n], ", ")); \\ Vaclav Kotesovec, Aug 21 2021

Scheme

(define (A226177 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) -2 0) (A226177 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

Formula

a(n) = mu(n)*d(n) = A008683(n)*A000005(n).

Sum_{n>0} a(n)/n^s = Product_{p prime} (1 - 2p^(-s)). - Ralf Stephan, Jul 07 2013

a(n) = mu(n) * 2^omega(n) = |mu(n)| * (-2)^omega(n), where omega = A001221. - Álvar Ibeas, Dec 30 2018

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

Extensions

More terms from Antti Karttunen, Jul 23 2017

Name changed by David A. Corneth, Jul 23 2017