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A347239 Sum of A347236 and its Dirichlet inverse.

%I A347239 #12 Dec 09 2021 01:31:08
%S A347239 2,0,0,1,0,4,0,13,4,4,0,26,0,8,8,55,0,34,0,26,16,4,0,26,4,8,68,52,0,0,
%T A347239 0,133,8,4,16,223,0,8,16,26,0,0,0,26,68,12,0,110,16,74,8,52,0,68,8,52,
%U A347239 16,4,0,4,0,12,136,463,16,0,0,26,24,0,0,247,0,8,148,52,16,0,0,110,421,4,0,8,8,8,8,26,0,8,32
%N A347239 Sum of A347236 and its Dirichlet inverse.
%C A347239 It seems that A030059 gives the positions of all zeros.
%H A347239 Antti Karttunen, <a href="/A347239/b347239.txt">Table of n, a(n) for n = 1..20000</a>
%H A347239 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A347239 a(n) = A347236(n) + A347238(n).
%F A347239 a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1<d<n} A347236(d) * A347238(n/d).
%F A347239 For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A347236(A030229(n)).
%F A347239 For all n >= 1, a(A001248(n)) = A000290(A001223(n)).
%o A347239 (PARI)
%o A347239 up_to = 16384;
%o A347239 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 A347239 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A347239 A061019(n) = (((-1)^bigomega(n))*n);
%o A347239 A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));
%o A347239 v347238 = DirInverseCorrect(vector(up_to,n,A347236(n)));
%o A347239 A347238(n) = v347238[n];
%o A347239 A347239(n) = (A347236(n)+A347238(n));
%Y A347239 Cf. A000290, A001223, A001248, A003961, A061019, A030059, A030229, A347236, A347238.
%K A347239 nonn
%O A347239 1,1
%A A347239 _Antti Karttunen_, Aug 24 2021