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%I
%S 1,1,1,-1,1,1,1,1,-1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,1,-1,1,1,-1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,1,1,-1,-1,-1,1,-1,1,1,1,1,1,1,1,-1,1,
%U 1,-1,-1,1,1,1,-1,1,1,1,-1,1,1,-1,-1,1,1,1,-1,-1,1,1,-1,1,1,1,1,1,-1,1,-1
%N Multiplicative function with a(p^e)=(-1)^(e-1)
%C a(A162644(n)) = +1; a(A162645(n)) = -1. [_Reinhard Zumkeller_, Jul 08 2009]
%H G. C. Greubel, <a href="/A162511/b162511.txt">Table of n, a(n) for n = 1..5000</a>
%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative functions</a>
%F Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
%F a(n) = (-1)^(A001222(n)-A001221(n)). - _Reinhard Zumkeller_, Jul 08 2009
%F a(n) = A076479(n) * A008836(n). - _R. J. Mathar_, Mar 30 2011
%e a(n)=1 when n is a squarefree number (A005117).
%p A162511 := proc(n)
%p local a,f;
%p a := 1;
%p for f in ifactors(n)[2] do
%p a := a*(-1)^(op(2,f)-1) ;
%p end do:
%p return a;
%p end proc: # _R. J. Mathar_, May 20 2017
%t a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* _Jean-François Alcover_, Apr 24 2017, after _Reinhard Zumkeller_ *)
%o (PARI) a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ _Charles R Greathouse IV_, Mar 09 2015
%o (Python)
%o from sympy import factorint
%o from operator import mul
%o def a(n):
%o f=factorint(n)
%o return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # _Indranil Ghosh_, May 20 2017
%Y Cf. A076479, A162510, A162512, A002035, A072587, A036537, A162643, A046660.
%K easy,mult,sign
%O 1,1
%A _Gerard P. Michon_, Jul 05 2009
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