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A347091 Sum of A332844 and its Dirichlet inverse.

%I A347091 #12 Aug 30 2021 21:51:27
%S A347091 2,0,0,9,0,24,0,21,16,36,0,28,0,48,48,37,0,36,0,42,64,72,0,60,36,84,
%T A347091 48,56,0,0,0,81,96,108,96,114,0,120,112,90,0,0,0,84,72,144,0,164,64,
%U A347091 84,144,98,0,120,144,120,160,180,0,216,0,192,96,166,168,0,0,126,192,0,0,258,0,228,112,140,192,0,0,246,132
%N A347091 Sum of A332844 and its Dirichlet inverse.
%C A347091 No negative terms in range 1 .. 2^20.
%C A347091 Apparently, A030059 gives the positions of all zeros.
%H A347091 Antti Karttunen, <a href="/A347091/b347091.txt">Table of n, a(n) for n = 1..16384</a>
%F A347091 a(n) = A332844(n) + A347090(n).
%F A347091 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A332844(d) * A347090(n/d).
%F A347091 For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A332844(A030229(n)).
%o A347091 (PARI)
%o A347091 up_to = 16384;
%o A347091 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 A347091 A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
%o A347091 v347090 = DirInverseCorrect(vector(up_to,n,A332844(n)));
%o A347091 A347090(n) = v347090[n];
%o A347091 A347091(n) = (A332844(n)+A347090(n));
%Y A347091 Cf. A030059, A030229, A332844, A347090.
%Y A347091 Cf. also A347093.
%K A347091 nonn
%O A347091 1,1
%A A347091 _Antti Karttunen_, Aug 18 2021