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%I #22 Oct 24 2023 04:37:23
%S 1,-3,-4,4,-6,12,-8,-6,6,18,-12,-16,-14,24,24,8,-18,-18,-20,-24,32,36,
%T -24,24,10,42,-12,-32,-30,-72,-32,-12,48,54,48,24,-38,60,56,36,-42,
%U -96,-44,-48,-36,72,-48,-32,14,-30,72,-56,-54,36,72,48,80,90,-60,96,-62,96,-48,16,84,-144,-68,-72,96,-144
%N Dirichlet inverse of A034448 (unitary sigma).
%H Andrew Howroyd, <a href="/A178450/b178450.txt">Table of n, a(n) for n = 1..1000</a>
%F Dirichlet g.f.: zeta(2s-1)/(zeta(s)*zeta(s-1)). - _R. J. Mathar_, Apr 14 2011
%F Multiplicative with a(p^e) = 2*p^(e/2) if e is even, -(p+1)*p^((e-1)/2) if e is odd. - _Sebastian Karlsson_, Dec 04 2021
%t usigma[n_] := If[n==1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
%t a[n_] := a[n] = If[n==1, 1, -Sum[usigma[n/d] a[d], {d, Most@Divisors[n]}]];
%t Array[a, 70] (* _Jean-François Alcover_, Feb 16 2020 *)
%t f[p_, e_] := If[OddQ[e], -(p+1)*p^((e-1)/2), 2*p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 24 2023 *)
%o (PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdivmult(n, d, if(gcd(d, n/d)==1, d))))} \\ _Andrew Howroyd_, Aug 05 2018
%o (Haskell)
%o import Math.NumberTheory.Primes
%o a n = product . map (\(p, e) -> if even e then 2*unPrime p^(e`div`2) else -(unPrime p+1)*unPrime p^(e`div`2)) $ factorise n -- _Sebastian Karlsson_, Dec 04 2021
%Y Cf. A034448.
%K sign,easy,mult
%O 1,2
%A _R. J. Mathar_, Dec 22 2010
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