A378756 -id:A378756 - 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

sign

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).

A379106 Dirichlet convolution of A000120 and A323910, where A323910 is the Dirichlet inverse of the deficiency of n, and A000120 is the binary weight of n.

Original entry on oeis.org

1, 0, 0, 0, -2, 2, -3, 0, -3, 2, -7, 4, -9, 2, 2, 0, -14, 4, -15, 4, -1, 2, -18, 8, -8, 2, -4, 4, -24, 2, -25, 0, -2, 2, 5, 12, -33, 2, 0, 8, -37, 0, -38, 4, 10, 2, -41, 16, -20, 0, 2, 4, -48, 2, 15, 8, 0, 2, -53, 12, -55, 2, 19, 0, 16, -8, -63, 4, -3, -4, -66, 32, -69, 2, -2, 4, 18, -12, -73, 16, -15, 2, -78, 8, 30

Offset: 1

Antti Karttunen, Dec 16 2024

sign

Cf. A000120, A033879, A323910, A379107 (Dirichlet inverse).

Cf. also A294898, A378754, A378756.

A033879(n) = (n+n-sigma(n));
memoA323910 = Map();
A323910(n) = if(1==n,1,my(v); if(mapisdefined(memoA323910,n,&v), v, v = -sumdiv(n,d,if(dA033879(n/d)*A323910(d),0)); mapput(memoA323910,n,v); (v)));
A379106(n) = sumdiv(n,d,hammingweight(d)*A323910(n/d));

a(n) = Sum_{d|n} A000120(d)*A323910(n/d).

cp's OEIS Frontend

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

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