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a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).
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%I #29 Sep 08 2022 08:45:05

%S 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,

%T 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,

%U -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

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

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

%C Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) [_R. J. Mathar_, Dec 15 2008]

%C Not multiplicative: For example a(2)*a(15) <> a(30). - _R. J. Mathar_, Mar 31 2012

%C Row products of table A225817. - _Reinhard Zumkeller_, Jul 30 2013

%H Reinhard Zumkeller, <a href="/A069158/b069158.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%p A069158 := proc(n)

%p mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;

%p end proc: # _R. J. Mathar_, May 28 2016

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

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

%o (Haskell)

%o a069158 = product . a225817_row -- _Reinhard Zumkeller_, Jul 30 2013

%K sign

%O 1,1

%A _Leroy Quet_, Apr 08 2002