A378990 Dirichlet inverse of the binary weight of n (A000120).

1, -1, -2, 0, -2, 2, -3, 0, 2, 2, -3, 0, -3, 3, 4, 0, -2, -2, -3, 0, 9, 3, -4, 0, 1, 3, -4, 0, -4, -4, -5, 0, 10, 2, 9, 0, -3, 3, 8, 0, -3, -9, -4, 0, -4, 4, -5, 0, 6, -1, 4, 0, -4, 4, 7, 0, 8, 4, -5, 0, -5, 5, -18, 0, 10, -10, -3, 0, 13, -9, -4, 0, -3, 3, 0, 0, 14, -8, -5, 0, 9, 3, -4, 0, 4, 4, 11, 0, -4, 4, 13, 0, 15

Offset: 1

Author

Antti Karttunen, Dec 15 2024

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to binary expansion of n

Crossrefs

Dirichlet inverse of A000120.

Möbius transform of A378989.

Cf. A008683, A378991.

Programs

PARI

memoA378990 = Map();
A378990(n) = if(1==n,1,my(v); if(mapisdefined(memoA378990,n,&v), v, v = -sumdiv(n,d,if(dA378990(d),0)); mapput(memoA378990,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA000120(n/d) * a(d).

a(n) = Sum_{d|n} A008683(d)*A378989(n/d).