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A178450 Dirichlet inverse of A034448 (unitary sigma).

%I A178450 #26 Nov 26 2024 17:14:13
%S A178450 1,-3,-4,4,-6,12,-8,-6,6,18,-12,-16,-14,24,24,8,-18,-18,-20,-24,32,36,
%T A178450 -24,24,10,42,-12,-32,-30,-72,-32,-12,48,54,48,24,-38,60,56,36,-42,
%U A178450 -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 A178450 Dirichlet inverse of A034448 (unitary sigma).
%H A178450 Antti Karttunen, <a href="/A178450/b178450.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Andrew Howroyd)
%F A178450 Dirichlet g.f.: zeta(2s-1)/(zeta(s)*zeta(s-1)). - _R. J. Mathar_, Apr 14 2011
%F A178450 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 A178450 usigma[n_] := If[n==1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
%t A178450 a[n_] := a[n] = If[n==1, 1, -Sum[usigma[n/d] a[d], {d, Most@Divisors[n]}]];
%t A178450 Array[a, 70] (* _Jean-François Alcover_, Feb 16 2020 *)
%t A178450 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 A178450 (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 A178450 (PARI) A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); }; \\ (After the multiplicative formula) - _Antti Karttunen_, Nov 26 2024
%o A178450 (Haskell)
%o A178450 import Math.NumberTheory.Primes
%o A178450 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 A178450 Cf. A034448, A378434.
%K A178450 sign,easy,mult
%O A178450 1,2
%A A178450 _R. J. Mathar_, Dec 22 2010