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A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

%I A347099 #10 Jan 24 2025 08:44:10
%S A347099 2,0,0,1,0,4,0,9,4,4,0,32,0,8,8,49,0,56,0,36,16,4,0,153,4,8,56,66,0,
%T A347099 96,0,207,8,4,16,295,0,8,16,187,0,168,0,48,120,12,0,553,16,80,8,78,0,
%U A347099 444,8,323,16,4,0,480,0,12,216,745,16,144,0,60,24,200,0,1016,0,8,152,90,16,216,0,723,472,4,0,786,8,8,8,289
%N A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).
%C A347099 Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.
%C A347099 The first negative term is a(720) = -6306.
%H A347099 Antti Karttunen, <a href="/A347099/b347099.txt">Table of n, a(n) for n = 1..20000</a>
%H A347099 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%H A347099 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A347099 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A336853(d) * A347098(n/d).
%F A347099 For all n >= 1, a(A001248(n)) = A001223(n)^2.
%o A347099 (PARI)
%o A347099 up_to = 16384;
%o A347099 A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
%o A347099 Aux347098(n) = if(1==n,n,A336853(n));
%o A347099 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 A347099 v347098 = DirInverseCorrect(vector(up_to,n,Aux347098(n)));
%o A347099 A347098(n) = v347098[n];
%o A347099 A347099(n) = if(1==n,2,A336853(n)+A347098(n));
%Y A347099 Cf. A001223, A001248, A003961, A063524, A336853, A347098.
%Y A347099 Cf. also A346250, A347097.
%K A347099 sign
%O A347099 1,1
%A A347099 _Antti Karttunen_, Aug 19 2021