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

1,2

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.

If n is semiprime, then a(n) = 4 * ( ceiling(sqrt(n)) - floor(sqrt(n)) )

= ( ceiling(sqrt(n)) - floor(sqrt(n)) )*( ceiling(sqrt(n)) - floor(sqrt(n)) + 3 ) = 2 - 2*(-1)^( ceiling(sqrt(n)) - floor(sqrt(n)) ).

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A350272 A171933 A074823 * A159916 A159286 A355837

Adjacent sequences: A226174 A226175 A226176 * A226178 A226179 A226180

Keyword

sign,mult

Author

Wesley Ivan Hurt, May 29 2013

Extensions

More terms from Antti Karttunen, Jul 23 2017

Name changed by David A. Corneth, Jul 23 2017

Status

approved