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Sum of A110963 and its Dirichlet inverse.
4

%I #11 Apr 18 2022 17:49:15

%S 2,0,0,1,0,2,0,1,1,4,0,1,0,2,4,1,0,5,0,2,2,4,0,1,4,8,5,1,0,-2,0,1,4,

%T 10,4,3,0,6,8,2,0,10,0,2,8,4,0,1,1,10,10,4,0,3,8,1,6,16,0,1,0,2,15,1,

%U 16,14,0,5,4,6,0,3,0,20,6,3,4,-2,0,2,9,22,0,6,20,12,16,2,0,16,8,2,2,4,12,1,0,25,24

%N Sum of A110963 and its Dirichlet inverse.

%C Note the negative terms, in contrast to A349135, which apparently has none.

%H Antti Karttunen, <a href="/A353367/b353367.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A110963(n) + A353366(n).

%F For n > 1, a(n) = -Sum_{d|n, 1<d<n} A110963(d) * A353366(n/d).

%F For all n >= 1, a(4*n) = A110963(n), and a(8*n-4) = A003602(n).

%o (PARI)

%o up_to = 65537;

%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 A003602(n) = (1+(n>>valuation(n,2)))/2;

%o A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));

%o v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));

%o A353366(n) = v353366[n];

%o A353367(n) = (A110963(n)+A353366(n));

%Y Cf. A003602, A110963, A353366.

%Y Cf. also A349135, A353369.

%K sign

%O 1,1

%A _Antti Karttunen_, Apr 18 2022