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A349629 Numerators of the Dirichlet inverse of the abundancy index, sigma(n)/n.

%I A349629 #14 Dec 01 2021 08:44:09
%S A349629 1,-3,-4,1,-6,2,-8,0,1,9,-12,-2,-14,12,8,0,-18,-1,-20,-3,32,18,-24,0,
%T A349629 1,21,0,-4,-30,-12,-32,0,16,27,48,1,-38,30,56,0,-42,-16,-44,-6,-2,36,
%U A349629 -48,0,1,-3,24,-7,-54,0,72,0,80,45,-60,4,-62,48,-8,0,84,-24,-68,-9,32,-72,-72,0,-74,57,-4,-10,96,-28
%N A349629 Numerators of the Dirichlet inverse of the abundancy index, sigma(n)/n.
%C A349629 Because the ratio A000203(n)/n [known as the abundancy index of n] is multiplicative, so is also its Dirichlet inverse. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349630 gives the denominators. See the examples.
%H A349629 Antti Karttunen, <a href="/A349629/b349629.txt">Table of n, a(n) for n = 1..20000</a>
%e A349629 The ratio a(n)/A349630(n) for n = 1..15: 1/1, -3/2, -4/3, 1/2, -6/5, 2/1, -8/7, 0/1, 1/3, 9/5, -12/11, -2/3, -14/13, 12/7, 8/5.
%t A349629 f[1] = 1; f[n_] := f[n] = -DivisorSum[n, f[#] * DivisorSigma[1, n/#] * #/n &, # < n &]; Numerator @ Array[f, 100] (* _Amiram Eldar_, Nov 28 2021 *)
%o A349629 (PARI)
%o A349629 up_to = 16384;
%o A349629 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A349629 Abi(n) = (sigma(n)/n);
%o A349629 vDirInv_of_Abi = DirInverseCorrect(vector(up_to,n,Abi(n)));
%o A349629 A349629(n) = numerator(vDirInv_of_Abi[n]);
%Y A349629 Cf. A000203, A017665, A017666.
%Y A349629 Cf. A349630 (denominators).
%K A349629 sign,frac
%O A349629 1,2
%A A349629 _Antti Karttunen_, Nov 27 2021