A347099 - OEIS

%I #6 Aug 20 2021 00:24:04

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

%C Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.

%C The first negative term is a(720) = -6306.

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A336853(d) * A347098(n/d).

%F For all n >= 1, a(A001248(n)) = A001223(n)^2.

%o (PARI)

%o up_to = 16384;

%o A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };

%o Aux347098(n) = if(1==n,n,A336853(n));

%o 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 v347098 = DirInverseCorrect(vector(up_to,n,Aux347098(n)));

%o A347098(n) = v347098[n];

%o A347099(n) = if(1==n,2,A336853(n)+A347098(n));

%Y Cf. A001223, A001248, A003961, A063524, A336853, A347098.

%Y Cf. also A346250, A347097.

%K sign

%O 1,1

%A _Antti Karttunen_, Aug 19 2021