A063441 - OEIS

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

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OFFSET

1,2

LINKS

Aloe Poliszuk, Table of n, a(n) for n = 1..10000 (First 2000 terms from Harry J. Smith)

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

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

a(n) = A048250(n) * mu(n) = sigma(rad(n)) * mu(n), where rad = A007947. - Aloe Poliszuk, Nov 04 2025

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI) a(n) = sumdiv(n, d, d*moebius(n));

(PARI) a(n) = direuler(p=2, n, 1-X-p*X)[n]; \\ Harry J. Smith, Aug 21 2009

CROSSREFS

Cf. A008683 (mu), A000203 (sigma), A048250, A055615.

Sequence in context: A105576 A105826 A110665 * A319600 A092894 A276563

Adjacent sequences: A063438 A063439 A063440 * A063442 A063443 A063444

KEYWORD

easy,sign,mult

AUTHOR

Jason Earls, Jul 23 2001

STATUS

approved