A353369 -id:A353369 - cp's OEIS frontend

A353367 Sum of A110963 and its Dirichlet inverse.

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, 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, 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

Offset: 1

Antti Karttunen, Apr 18 2022

sign

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

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003602, A110963, A353366.

Cf. also A349135, A353369.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];
A353367(n) = (A110963(n)+A353366(n));

a(n) = A110963(n) + A353366(n).
For n > 1, a(n) = -Sum_{d|n, 1A110963(d) * A353366(n/d).
For all n >= 1, a(4*n) = A110963(n), and a(8*n-4) = A003602(n).

A353368 Dirichlet inverse of A103391, "even fractal sequence".

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -3, -1, 2, 2, -4, -3, -3, 4, 3, 0, -2, -10, -6, 1, 8, 4, -7, 3, 1, -2, -8, -1, -5, -4, -9, -1, 14, -10, 2, 17, -6, 4, 1, -1, -4, -22, -12, 1, -3, 4, -13, -1, 6, -14, -6, 11, -8, 28, 1, 1, 19, -10, -16, 3, -9, 4, -25, -1, 10, -42, -18, 25, 18, 0, -19, -17, -6, -14, -12, 5, 13, 12, -21, 3, 24

Offset: 1

Antti Karttunen, Apr 18 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003602, A103391, A353369.

Cf. also A349134, A353366.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1)));
v353368 = DirInverseCorrect(vector(up_to,n,A103391(n)));
A353368(n) = v353368[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A103391(n/d) * a(d).

a(n) = A353369(n) - A103391(n).

cp's OEIS Frontend

A353367 Sum of A110963 and its Dirichlet inverse.

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