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A347094 Sum of A038040 (convolution of sigma with Euler phi) and its Dirichlet inverse.

%I A347094 #14 Aug 30 2021 21:51:54
%S A347094 2,0,0,16,0,48,0,32,36,80,0,48,0,112,120,80,0,72,0,80,168,176,0,192,
%T A347094 100,208,108,112,0,0,0,192,264,272,280,360,0,304,312,320,0,0,0,176,
%U A347094 180,368,0,480,196,200,408,208,0,432,440,448,456,464,0,960,0,496,252,448,520,0,0,272,552,0,0,864,0,592,300,304,616
%N A347094 Sum of A038040 (convolution of sigma with Euler phi) and its Dirichlet inverse.
%C A347094 No negative terms in range 1 .. 2^20.
%C A347094 Apparently, A030059 gives the positions of all zeros.
%H A347094 Antti Karttunen, <a href="/A347094/b347094.txt">Table of n, a(n) for n = 1..16384</a>
%F A347094 a(n) = A038040(n) + A328722(n).
%F A347094 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A038040(d) * A328722(n/d).
%F A347094 For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A038040(A030229(n)).
%o A347094 (PARI)
%o A347094 up_to = 16384;
%o A347094 A038040(n) = (n*numdiv(n));
%o A347094 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 A347094 v328722 = DirInverseCorrect(vector(up_to,n,A038040(n)));
%o A347094 A328722(n) = v328722[n];
%o A347094 A347094(n) = (A038040(n)+A328722(n));
%Y A347094 Cf. A000005, A030059, A030229, A038040, A328722, A347093.
%K A347094 nonn
%O A347094 1,1
%A A347094 _Antti Karttunen_, Aug 18 2021