A063441 a(n) = sigma(n) * mu(n).

1, -3, -4, 0, -6, 12, -8, 0, 0, 18, -12, 0, -14, 24, 24, 0, -18, 0, -20, 0, 32, 36, -24, 0, 0, 42, 0, 0, -30, -72, -32, 0, 48, 54, 48, 0, -38, 60, 56, 0, -42, -96, -44, 0, 0, 72, -48, 0, 0, 0, 72, 0, -54, 0, 72, 0, 80, 90, -60, 0, -62, 96, 0, 0, 84, -144, -68, 0, 96, -144, -72, 0, -74, 114, 0, 0, 96, -168, -80, 0, 0, 126, -84, 0, 108

Offset: 1

Author

Jason Earls, Jul 23 2001

Keywords

• easy
• sign
• mult

Examples

			n=6: divisors of 6 are = [1, 2, 3, 6] and 1 * mu(6) + 2 * mu(6) + 3 * mu(6) + 6 * mu(6) = 12.
		

Links

• Harry J. Smith, Table of n, a(n) for n=1..2000

Programs

Mathematica

a[n_] := DivisorSigma[1, n] MoebiusMu[n]; Array[a, 90] (* Jean-François Alcover, Dec 05 2015 *)

PARI

j=[]; for(n=1,200,j=concat(j,sumdiv(n,d,d*moebius(n)))); j

PARI

a(n)=if(n<1,0,direuler(p=2,n,1-X-p*X)[n]) /* Ralf Stephan, Jul 07 2013 */

PARI

{ for (n=1, 2000, write("b063441.txt", n, " ", direuler(p=2, n, 1-X-p*X)[n]) ) } \\ Harry J. Smith, Aug 21 2009

Formula

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

a(n) = A000203(n) * A008683(n).

a(n) = A003959(n) * A008683(n) if n is squarefree, 0 otherwise. - Ralf Stephan, Mar 26 2004

Multiplicative with a(p^e) = -p-1, if e = 1, 0 otherwise. - Mitch Harris, Jun 27 2005, sign flipped by R. J. Mathar, May 29 2011

sum(n>0, a(n)/n^s) = product(p prime, 1-p^(-s)-p^(1-s) ). - Ralf Stephan, Jul 07 2013