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A347229 Sum of A344695 [gcd(sigma(n), psi(n))] and its Dirichlet inverse.

%I A347229 #14 Aug 30 2021 21:52:46
%S A347229 2,0,0,9,0,24,0,-21,16,36,0,-28,0,48,48,73,0,-42,0,-42,64,72,0,108,36,
%T A347229 84,-56,-56,0,0,0,-213,96,108,96,121,0,120,112,162,0,0,0,-84,-84,144,
%U A347229 0,-284,64,-102,144,-98,0,192,144,216,160,180,0,216,0,192,-112,649,168,0,0,-126,192,0,0,-357,0,228,-136
%N A347229 Sum of A344695 [gcd(sigma(n), psi(n))] and its Dirichlet inverse.
%C A347229 It seems that A030059 gives the positions of all zeros.
%H A347229 Antti Karttunen, <a href="/A347229/b347229.txt">Table of n, a(n) for n = 1..12000</a>
%F A347229 a(n) = A344695(n) + A347228(n).
%F A347229 a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1<d<n} A344695(d) * A347228(n/d).
%F A347229 For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A344695(A030229(n)). [Even though A344695 is not multiplicative, this holds because on squarefree n it is equal to psi(n) and sigma(n) that are multiplicative functions]
%o A347229 (PARI)
%o A347229 up_to = 16384;
%o A347229 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 A347229 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 A347229 A344695(n) = gcd(sigma(n), A001615(n));
%o A347229 v347228 = DirInverseCorrect(vector(up_to,n,A344695(n)));
%o A347229 A347228(n) = v347228[n];
%o A347229 A347229(n) = (A344695(n)+A347228(n));
%Y A347229 Cf. A000203, A001615, A030059, A030229, A344695, A347228, A347231.
%K A347229 sign
%O A347229 1,1
%A A347229 _Antti Karttunen_, Aug 25 2021