A378757 -id:A378757 - cp's OEIS frontend

A378756 Dirichlet convolution of A046692 and A005187, where A046692 is the Dirichlet inverse of sigma.

1, 0, 0, 0, 2, -2, 3, 0, 3, -6, 7, 0, 9, -8, -6, 0, 14, -6, 15, 0, -5, -16, 18, 0, 4, -20, -2, 0, 24, 14, 25, 0, -12, -30, -15, 0, 33, -32, -18, 0, 37, 12, 38, 0, -12, -38, 41, 0, 14, -8, -30, 0, 48, 4, -33, 0, -30, -50, 53, 0, 55, -52, -27, 0, -38, 26, 63, 0, -33, 32, 66, 0, 69, -68, -6, 0, -38, 38, 73, 0, 7, -76, 78

Offset: 1

Antti Karttunen, Dec 15 2024

sign

Cf. A005187, A046692, A378757 (Dirichlet inverse).

Cf. also A294898.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
A378756(n) = sumdiv(n,d,A046692(d)*A005187(n/d));

a(n) = Sum_{d|n} A046692(d)*A005187(n/d).

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.

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, 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).

cp's OEIS Frontend

A378756 Dirichlet convolution of A046692 and A005187, where A046692 is the Dirichlet inverse of sigma.

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