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A349630 Denominators of the Dirichlet inverse of the abundancy index, sigma(n)/n.

%I A349630 #9 Nov 28 2021 17:08:03
%S A349630 1,2,3,2,5,1,7,1,3,5,11,3,13,7,5,1,17,2,19,5,21,11,23,1,5,13,1,7,29,5,
%T A349630 31,1,11,17,35,6,37,19,39,1,41,7,43,11,5,23,47,1,7,10,17,13,53,1,55,1,
%U A349630 57,29,59,5,61,31,21,1,65,11,67,17,23,35,71,1,73,37,15,19,77,13,79,1,1,41,83,21,85,43,29,1,89
%N A349630 Denominators of the Dirichlet inverse of the abundancy index, sigma(n)/n.
%H A349630 Antti Karttunen, <a href="/A349630/b349630.txt">Table of n, a(n) for n = 1..20000</a>
%t A349630 f[1] = 1; f[n_] := f[n] = -DivisorSum[n, f[#] * DivisorSigma[1, n/#] * #/n &, # < n &]; Denominator @ Array[f, 100] (* _Amiram Eldar_, Nov 28 2021 *)
%o A349630 (PARI)
%o A349630 up_to = 16384;
%o A349630 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, correctly!
%o A349630 Abi(n) = (sigma(n)/n);
%o A349630 vDirInv_of_Abi = DirInverseCorrect(vector(up_to,n,Abi(n)));
%o A349630 A349630(n) = denominator(vDirInv_of_Abi[n]);
%Y A349630 Cf. A000203, A017665, A017666.
%Y A349630 Cf. A349629 (numerators).
%K A349630 nonn,frac
%O A349630 1,2
%A A349630 _Antti Karttunen_, Nov 27 2021