A349344 Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.

1, -2, -2, 0, -3, 4, -4, 0, -1, 6, -6, 0, -7, 8, 4, 0, -9, 2, -10, 0, 5, 12, -12, 0, -4, 14, -2, 0, -15, -8, -16, 0, 7, 18, 6, 0, -19, 20, 8, 0, -21, -10, -22, 0, 3, 24, -24, 0, -9, 8, 10, 0, -27, 4, 8, 0, 11, 30, -30, 0, -31, 32, 4, 0, 9, -14, -34, 0, 13, -12, -36, 0, -37, 38, 8, 0, 9, -16, -40, 0, -4, 42, -42, 0

Offset: 1

Author

Antti Karttunen, Nov 15 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A109168 (A140472), A349345.

Cf. also A328203, A349134, A349343, A349346, A349353.

Programs

PARI

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA109168(n) = ((n+bitand(n, -n))\2); \\ From A109168 by M. F. Hasler, Oct 19 2019 (Cf. A140472).
v349344 = DirInverseCorrect(vector(up_to,n,A109168(n)));
A349344(n) = v349344[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A109168(n/d) * a(d).

a(n) = A349345(n) - A109168(n).