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

%I A069158 #36 Feb 10 2025 01:37:28
%S A069158 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 A069158 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 A069158 -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 A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).
%C A069158 Absolute value of a(n) = absolute value of mu(n).
%C A069158 Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) - _R. J. Mathar_, Dec 15 2008
%C A069158 Not multiplicative: For example a(2)*a(15) <> a(30). - _R. J. Mathar_, Mar 31 2012
%C A069158 Row products of table A225817. - _Reinhard Zumkeller_, Jul 30 2013
%H A069158 Reinhard Zumkeller, <a href="/A069158/b069158.txt">Table of n, a(n) for n = 1..10000</a>
%F A069158 a(n) = 0 if mu(n) = 0 (A013929); a(n) = -1 if n = prime; a(n) = 1 if n = squarefree composite (A120944) or 1.
%F A069158 a(n) = A008966(n) - 2*A010051(n). - _Amiram Eldar_, Feb 10 2025
%e A069158 a(6) = mu(1)*mu(2)*mu(3)*mu(6) = 1*(-1)*(-1)*1 = 1.
%p A069158 A069158 := proc(n)
%p A069158     mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;
%p A069158 end proc: # _R. J. Mathar_, May 28 2016
%t A069158 a[n_] := Product[MoebiusMu[d], {d, Divisors[n]}]; Array[a, 106] (* _Jean-François Alcover_, Feb 22 2018 *)
%o A069158 (Magma) f := function(n); t1 := &*[MoebiusMu(d) : d in Divisors(n) ]; return t1; end function;
%o A069158 (Haskell)
%o A069158 a069158 = product . a225817_row  -- _Reinhard Zumkeller_, Jul 30 2013
%o A069158 (PARI) a(n) = vecprod(apply(moebius, divisors(n))); \\ _Amiram Eldar_, Feb 10 2025
%Y A069158 Cf. A008683, A008966, A010051, A013929, A046389, A046391, A080323, A225817.
%K A069158 sign
%O A069158 1,1
%A A069158 _Leroy Quet_, Apr 08 2002