A378989 Dirichlet inverse of the Möbius transform of binary weight of n.

1, 0, -1, 0, -1, 0, -2, 0, 1, 0, -2, 0, -2, 0, 1, 0, -1, 0, -2, 0, 5, 0, -3, 0, 0, 0, -3, 0, -3, 0, -4, 0, 6, 0, 5, 0, -2, 0, 4, 0, -2, 0, -3, 0, -1, 0, -4, 0, 4, 0, 1, 0, -3, 0, 3, 0, 4, 0, -4, 0, -4, 0, -11, 0, 6, 0, -2, 0, 8, 0, -3, 0, -2, 0, 2, 0, 9, 0, -4, 0, 6, 0, -3, 0, 1, 0, 6, 0, -3, 0, 8, 0, 9, 0, 2, 0, -2, 0, -12, 0, -3, 0, -4, 0, -12

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 A297115.

Inverse Möbius transform of A378990.

Cf. A000120.

Programs

PARI

A297115(n) = sumdiv(n, d, moebius(n/d)*hammingweight(d));
memoA378989 = Map();
A378989(n) = if(1==n,1,my(v); if(mapisdefined(memoA378989,n,&v), v, v = -sumdiv(n,d,if(dA297115(n/d)*A378989(d),0)); mapput(memoA378989,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA297115(n/d) * a(d).

a(n) = Sum_{d|n} A378990(d).