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A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.

%I A347095 #11 Aug 30 2021 21:52:03
%S A347095 2,0,0,9,0,30,0,21,25,54,0,35,0,78,90,49,0,51,0,63,130,126,0,95,81,
%T A347095 150,85,91,0,0,0,113,210,198,234,172,0,222,250,171,0,0,0,147,153,270,
%U A347095 0,235,169,147,330,175,0,231,378,247,370,342,0,405,0,366,221,257,450,0,0,231,450,0,0,424,0,438,245,259,546
%N A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.
%C A347095 No negative terms in range 1 .. 2^20.
%C A347095 Apparently, A030059 gives the positions of all zeros.
%H A347095 Antti Karttunen, <a href="/A347095/b347095.txt">Table of n, a(n) for n = 1..16384</a>
%F A347095 a(n) = A018804(n) + A101035(n).
%F A347095 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A018804(d) * A101035(n/d).
%F A347095 For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A018804(A030229(n)).
%o A347095 (PARI)
%o A347095 up_to = 16384;
%o A347095 A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
%o A347095 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 A347095 v101035 = DirInverseCorrect(vector(up_to,n,A018804(n)));
%o A347095 A101035(n) = v101035[n];
%o A347095 A347095(n) = (A018804(n)+A101035(n));
%Y A347095 Cf. A018804, A030059, A030229, A101035.
%Y A347095 Cf. also A347094.
%K A347095 nonn
%O A347095 1,1
%A A347095 _Antti Karttunen_, Aug 18 2021