A323399 -id:A323399 - cp's OEIS frontend

A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

2, 0, 0, 9, 0, 24, 0, 9, 16, 36, 0, 12, 0, 48, 48, 27, 0, 24, 0, 18, 64, 72, 0, 60, 36, 84, 32, 24, 0, 0, 0, 45, 96, 108, 96, 84, 0, 120, 112, 90, 0, 0, 0, 36, 48, 144, 0, 84, 64, 72, 144, 42, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 64, 99, 168, 0, 0, 54, 192, 0, 0, 132, 0, 228, 96, 60, 192, 0, 0, 126, 112, 252, 0, 288, 216, 264, 240, 180, 0

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

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

Cf. A001615, A323363.

Cf. also A319340, A322581, A323365, A323372, A323399.

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));
A323364(n) = (A001615(n) + A323363(n));

A323403 Sum of sigma and its Dirichlet inverse: a(n) = A000203(n) + A046692(n).

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 15, 16, 36, 0, 20, 0, 48, 48, 31, 0, 30, 0, 30, 64, 72, 0, 60, 36, 84, 40, 40, 0, 0, 0, 63, 96, 108, 96, 97, 0, 120, 112, 90, 0, 0, 0, 60, 60, 144, 0, 124, 64, 78, 144, 70, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 80, 127, 168, 0, 0, 90, 192, 0, 0, 195, 0, 228, 104, 100, 192, 0, 0, 186, 121, 252, 0, 288, 216, 264, 240, 180, 0, 288

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A046692.

Cf. also A319340, A322581, A323364, A323399, A323408.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v046692 = DirInverse(vector(up_to,n,sigma(n)));
A046692(n) = v046692[n];
A323403(n) = (sigma(n)+A046692(n));

a(n) = A000203(n) + A046692(n).

A323408 Sum of unitary phi and its Dirichlet inverse: a(n) = A047994(n) + A323407(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 5, 4, 8, 0, 10, 0, 12, 16, 15, 0, 12, 0, 20, 24, 20, 0, 18, 16, 24, 24, 30, 0, 0, 0, 35, 40, 32, 48, 32, 0, 36, 48, 36, 0, 0, 0, 50, 48, 44, 0, 30, 36, 32, 64, 60, 0, 28, 80, 54, 72, 56, 0, 8, 0, 60, 72, 71, 96, 0, 0, 80, 88, 0, 0, 64, 0, 72, 64, 90, 120, 0, 0, 60, 88, 80, 0, 12, 128, 84, 112, 90, 0, 16, 144, 110, 120

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

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

Cf. A047994, A323407.

Cf. also A319340, A322581, A323364, A323399, A323403.

up_to = 65537;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v323407 = DirInverse(vector(up_to,n,A047994(n)));
A323407(n) = v323407[n];
A323408(n) = (A047994(n) + A323407(n));

cp's OEIS Frontend

A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

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A323403 Sum of sigma and its Dirichlet inverse: a(n) = A000203(n) + A046692(n).

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A323408 Sum of unitary phi and its Dirichlet inverse: a(n) = A047994(n) + A323407(n).

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PARI