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%I #15 May 21 2017 21:32:43
%S 1,-2,-2,2,-2,4,-2,-2,2,4,-2,-4,-2,4,4,2,-2,-4,-2,-4,4,4,-2,4,2,4,-2,
%T -4,-2,-8,-2,-2,4,4,4,4,-2,4,4,4,-2,-8,-2,-4,-4,4,-2,-4,2,-4,4,-4,-2,
%U 4,4,4,4,4,-2,8,-2,4,-4,2,4,-8,-2,-4,4,-8,-2,-4,-2,4,-4,-4,4,-8,-2,-4,2,4,-2
%N Dirichlet inverse of number of unitary divisors of n (A034444).
%C Abs{a(n)} = A034444(n). Examples of Dirichlet convolutions with function a(n), i.e. b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A034444(n) = A063524(n), a(n) * A000005(n) = A010052(n), a(n) * A000027(n) = A074722(n), a(n) * A000012(n) = A008836(n).
%H G. C. Greubel, <a href="/A158522/b158522.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = (-1)^A001222(n)*A034444(n) = (-1)^A001222(n)*2^A001221(n), for n >= 2.
%F Multiplicative with a(p^e) = 2*(-1)^e, p prime, e>0. a(p^0) = 1.
%F Dirichlet g.f.: zeta(2s)/(zeta(s))^2. - _R. J. Mathar_, Apr 02 2011
%e a(60) = a(2^2*3*5) = [(-1)^2*2]*[(-1)^1*2]*[(-1)^1*2] = 2*(-2)*(-2) = 8.
%t Table[LiouvilleLambda[n] 2^PrimeNu[n], {n, 1, 50}] (* _Geoffrey Critzer_, Mar 07 2015 *)
%o (PARI) for(n=1,20, print1((-1)^bigomega(n)* 2^omega(n), ", ")) \\ _G. C. Greubel_, May 21 2017
%Y Cf. A034444, A063524, A000005, A010052, A000027, A074722, A001222, A001221, A000040, A006881, A120944, A000961.
%K sign,mult
%O 1,2
%A _Jaroslav Krizek_, Mar 20 2009
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