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Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.
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%I #14 Aug 30 2021 21:51:42

%S 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 100,208,96,84,0,0,0,156,264,272,280,336,0,304,312,320,0,0,0,132,160,

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

%C No negative terms in range 1 .. 2^20.

%C Apparently, A030059 gives the positions of all zeros.

%H Antti Karttunen, <a href="/A347093/b347093.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A322577(n) + A347092(n).

%F For n > 1, a(n) = -Sum_{d|n, 1<d<n} A322577(d) * A347092(n/d).

%F For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A322577(A030229(n)).

%o (PARI)

%o up_to = 16384;

%o 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 A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));

%o 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 v347092 = DirInverseCorrect(vector(up_to,n,A322577(n)));

%o A347092(n) = v347092[n];

%o A347093(n) = (A322577(n)+A347092(n));

%Y Cf. A000010, A001615, A030059, A030229, A322577, A347092, A347094.

%K nonn

%O 1,1

%A _Antti Karttunen_, Aug 18 2021