A378991 Dirichlet inverse of the Möbius transform of A005187, where A005187(n) = 2*n - (number of 1's in binary representation of n).

1, -2, -3, 0, -7, 8, -10, 0, -3, 20, -18, -4, -22, 28, 27, 0, -31, 6, -34, -12, 35, 52, -41, 0, 10, 64, 11, -16, -53, -104, -56, 0, 66, 92, 91, 4, -70, 100, 84, 0, -78, -132, -81, -32, 21, 120, -88, 0, 16, -66, 123, -40, -101, -56, 173, 0, 132, 156, -112, 124, -116, 164, 51, 0, 210, -256, -130, -60, 156, -364, -137

Offset: 1

Author

Antti Karttunen, Dec 15 2024

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20020

Crossrefs

Dirichlet inverse of A297111.

Inverse Möbius transform of A346237.

Cf. A005187.

Cf. also A378989, A378990.

Programs

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));
memoA378991 = Map();
A378991(n) = if(1==n,1,my(v); if(mapisdefined(memoA378991,n,&v), v, v = -sumdiv(n,d,if(dA297111(n/d)*A378991(d),0)); mapput(memoA378991,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA297111(n/d) * a(d).

a(n) = Sum_{d|n} A346237(d).