A378989 -id:A378989 - cp's OEIS frontend

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

Antti Karttunen, Dec 15 2024

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

Dirichlet inverse of A000120.
Möbius transform of A378989.
Cf. A008683, A378991.

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)));

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).

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

Original entry on oeis.org

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

Antti Karttunen, Dec 15 2024

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

Dirichlet inverse of A297111.
Inverse Möbius transform of A346237.
Cf. A005187.
Cf. also A378989, A378990.

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)));

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

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

cp's OEIS Frontend

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

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