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A349337 Dirichlet inverse of A230593.

%I A349337 #13 Nov 21 2021 01:17:59
%S A349337 1,-3,-4,3,-6,13,-8,-3,4,19,-12,-16,-14,25,25,3,-18,-17,-20,-22,33,37,
%T A349337 -24,19,6,43,-4,-28,-30,-87,-32,-3,49,55,49,33,-38,61,57,25,-42,-113,
%U A349337 -44,-40,-29,73,-48,-22,8,-25,73,-46,-54,21,73,31,81,91,-60,125,-62,97,-37,3,85,-165,-68,-58,97,-163,-72,-52
%N A349337 Dirichlet inverse of A230593.
%C A349337 Coincides with A347084 on all squarefree numbers (A005117), but also on n=81, where a(81) = A347084(81) = 4. Question: Are there any other such numbers?
%H A349337 Antti Karttunen, <a href="/A349337/b349337.txt">Table of n, a(n) for n = 1..20000</a>
%F A349337 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A230593(d) * A349337(n/d).
%o A349337 (PARI)
%o A349337 up_to = 20000;
%o A349337 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 A349337 A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
%o A349337 v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
%o A349337 A349337(n) = v349337[n];
%Y A349337 Cf. A005117, A069359, A230593.
%Y A349337 Cf. also A347084, A349434, A349435.
%K A349337 sign
%O A349337 1,2
%A A349337 _Antti Karttunen_, Nov 15 2021