A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 1, 1

Offset: 1

Author

Leroy Quet, Apr 08 2002

Keywords

• sign

Comments

Absolute value of a(n) = absolute value of mu(n).

Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) - R. J. Mathar, Dec 15 2008

Not multiplicative: For example a(2)*a(15) <> a(30). - R. J. Mathar, Mar 31 2012

Row products of table A225817. - Reinhard Zumkeller, Jul 30 2013

Examples

			a(6) = mu(1)*mu(2)*mu(3)*mu(6) = 1*(-1)*(-1)*1 = 1.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008683, A008966, A010051, A013929, A046389, A046391, A080323, A225817.

Programs

Haskell

a069158 = product . a225817_row  -- Reinhard Zumkeller, Jul 30 2013

Magma

f := function(n); t1 := &*[MoebiusMu(d) : d in Divisors(n) ]; return t1; end function;

Maple

A069158 := proc(n)
    mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, May 28 2016

Mathematica

a[n_] := Product[MoebiusMu[d], {d, Divisors[n]}]; Array[a, 106] (* Jean-François Alcover, Feb 22 2018 *)

PARI

a(n) = vecprod(apply(moebius, divisors(n))); \\ Amiram Eldar, Feb 10 2025

Formula

a(n) = 0 if mu(n) = 0 (A013929); a(n) = -1 if n = prime; a(n) = 1 if n = squarefree composite (A120944) or 1.

a(n) = A008966(n) - 2*A010051(n). - Amiram Eldar, Feb 10 2025