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A347093 Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.

%I A347093 #14 Aug 30 2021 21:51:42
%S A347093 2,0,0,16,0,48,0,24,36,80,0,36,0,112,120,73,0,64,0,60,168,176,0,192,
%T A347093 100,208,96,84,0,0,0,156,264,272,280,336,0,304,312,320,0,0,0,132,160,
%U A347093 368,0,378,196,192,408,156,0,432,440,448,456,464,0,960,0,496,224,373,520,0,0,204,552,0,0,688,0,592,288,228,616
%N A347093 Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.
%C A347093 No negative terms in range 1 .. 2^20.
%C A347093 Apparently, A030059 gives the positions of all zeros.
%H A347093 Antti Karttunen, <a href="/A347093/b347093.txt">Table of n, a(n) for n = 1..16384</a>
%F A347093 a(n) = A322577(n) + A347092(n).
%F A347093 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A322577(d) * A347092(n/d).
%F A347093 For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A322577(A030229(n)).
%o A347093 (PARI)
%o A347093 up_to = 16384;
%o A347093 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A347093 A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
%o A347093 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 A347093 v347092 = DirInverseCorrect(vector(up_to,n,A322577(n)));
%o A347093 A347092(n) = v347092[n];
%o A347093 A347093(n) = (A322577(n)+A347092(n));
%Y A347093 Cf. A000010, A001615, A030059, A030229, A322577, A347092, A347094.
%K A347093 nonn
%O A347093 1,1
%A A347093 _Antti Karttunen_, Aug 18 2021