A346237 -id:A346237 - cp's OEIS frontend

A378757 Dirichlet convolution of sigma and the Dirichlet inverse of A005187 (A346237).

1, 0, 0, 0, -2, 2, -3, 0, -3, 6, -7, 0, -9, 8, 6, 0, -14, 6, -15, 0, 5, 16, -18, 0, 0, 20, 2, 0, -24, -22, -25, 0, 12, 30, 27, 4, -33, 32, 18, 0, -37, -24, -38, 0, 24, 38, -41, 0, -5, -16, 30, 0, -48, -16, 61, 0, 30, 50, -53, 24, -55, 52, 45, 0, 74, -54, -63, 0, 33, -100, -66, 0, -69, 68, -18, 0, 80, -74, -73, 0, 2, 76, -78

Offset: 1

Antti Karttunen, Dec 15 2024

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Cf. A000203, A005187, A346237, A378756 (Dirichlet inverse).

Cf. also A294898.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
memoA346237 = Map();
A346237(n) = if(1==n,1,my(v); if(mapisdefined(memoA346237,n,&v), v, v = -sumdiv(n,d,if(dA005187(n/d)*A346237(d),0)); mapput(memoA346237,n,v); (v)));
A378757(n) = sumdiv(n,d,sigma(d)*A346237(n/d));

a(n) = Sum_{d|n} A000203(d)*A346237(n/d).

A346238 Sum of A005187 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 15, 16, 48, 0, 8, 0, 66, 64, 31, 0, 32, 0, 4, 88, 114, 0, 50, 64, 138, 64, 7, 0, -116, 0, 63, 152, 192, 176, 80, 0, 210, 184, 90, 0, -138, 0, -1, 80, 252, 0, 94, 121, -6, 256, -5, 0, 28, 304, 125, 280, 324, 0, 390, 0, 342, 136, 127, 368, -276, 0, -20, 336, -386, 0, 126, 0, 426, 24, -17, 418, -360, 0

Offset: 1

Antti Karttunen, Jul 13 2021

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Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A346237.

Cf. also A300244.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
v346237 = DirInverseCorrect(vector(up_to,n,A005187(n)));
A346237(n) = v346237[n];
A346238(n) = (A005187(n)+A346237(n));

a(n) = A005187(n) + A346237(n).

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

A378757 Dirichlet convolution of sigma and the Dirichlet inverse of A005187 (A346237).

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A346238 Sum of A005187 and its Dirichlet inverse.

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

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