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A349616 Dirichlet convolution of A000027 (the identity function) with the Dirichlet inverse of the inverse permutation of EKG-permutation.

%I A349616 #7 Nov 25 2021 19:19:42
%S A349616 1,0,-2,1,-5,6,-7,-2,13,11,-9,-6,-15,15,49,0,-16,-42,-18,-15,69,21,
%T A349616 -20,24,51,29,-48,-21,-28,-168,-30,-1,97,34,150,65,-30,38,141,48,-33,
%U A349616 -236,-38,-32,-317,44,-42,-40,97,-163,163,-36,-47,248,192,75,183,58,-48,294,-54,62,-443,1,301,-338,-61,-50,211
%N A349616 Dirichlet convolution of A000027 (the identity function) with the Dirichlet inverse of the inverse permutation of EKG-permutation.
%H A349616 Antti Karttunen, <a href="/A349616/b349616.txt">Table of n, a(n) for n = 1..20000</a>
%H A349616 <a href="/index/Ed#EKG">Index entries for sequences related to EKG sequence</a>
%F A349616 a(n) = Sum_{d|n} d * A323411(n/d).
%o A349616 (PARI)
%o A349616 up_to = 32768;
%o A349616 v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ Data prepared with _Chai Wah Wu_'s Dec 08 2014 Python-program given in A064413.
%o A349616 A064413(n) = v064413[n];
%o A349616 \\ Then its inverse A064664 was prepared:
%o A349616 m064664 = Map();
%o A349616 for(n=1,65539,mapput(m064664,A064413(n),n));
%o A349616 A064664(n) = mapget(m064664,n);
%o A349616 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 A349616 v323411 = DirInverseCorrect(vector(up_to,n,A064664(n)));
%o A349616 A323411(n) = v323411[n];
%o A349616 A349616(n) = sumdiv(n,d,d*A323411(n/d));
%Y A349616 Cf. A000027, A064413, A064664, A323411, A349617 (Dirichlet inverse).
%Y A349616 Cf. also A349613, A349614.
%K A349616 sign
%O A349616 1,3
%A A349616 _Antti Karttunen_, Nov 23 2021