A378990 -id:A378990 - cp's OEIS frontend

A379107 Dirichlet convolution of A033879 and A378990, where A033879 is the deficiency of n, and A378990 is the Dirichlet inverse of the binary weight of n.

1, 0, 0, 0, 2, -2, 3, 0, 3, -2, 7, -4, 9, -2, -2, 0, 14, -4, 15, -4, 1, -2, 18, -8, 12, -2, 4, -4, 24, -10, 25, 0, 2, -2, 7, -8, 33, -2, 0, -8, 37, -12, 38, -4, 2, -2, 41, -16, 29, -8, -2, -4, 48, -14, 13, -8, 0, -2, 53, -20, 55, -2, -1, 0, 20, -20, 63, -4, 3, -16, 66, -16, 69, -2, -6, -4, 24, -24, 73, -16, 24, -2, 78

Offset: 1

Antti Karttunen, Dec 16 2024

sign

Cf. A000120, A033879, A378990, A379106 (Dirichlet inverse).

Cf. also A294898, A378755, A378757.

A033879(n) = (n+n-sigma(n));
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)));
A379107(n) = sumdiv(n,d,A033879(d)*A378990(n/d));

a(n) = Sum_{d|n} A033879(d)*A378990(n/d).

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

Original entry on oeis.org

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

Antti Karttunen, Dec 15 2024

sign

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

Dirichlet inverse of A297115.
Inverse Möbius transform of A378990.
Cf. A000120.

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)));

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

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

sign

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

A379107 Dirichlet convolution of A033879 and A378990, where A033879 is the deficiency of n, and A378990 is the Dirichlet inverse of the binary weight of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula