A166633 -id:A166633 - cp's OEIS frontend

A166634 Totally multiplicative sequence with a(p) = 4*(p-1) for prime p.

1, 4, 8, 16, 16, 32, 24, 64, 64, 64, 40, 128, 48, 96, 128, 256, 64, 256, 72, 256, 192, 160, 88, 512, 256, 192, 512, 384, 112, 512, 120, 1024, 320, 256, 384, 1024, 144, 288, 384, 1024, 160, 768, 168, 640, 1024, 352, 184, 2048, 576, 1024

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A003958, A165825, A166633.

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] :=
DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*4^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (4*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = 4*(f[k,1]-1)); factorback(f);} \\ Michel Marcus, May 20 2016

Multiplicative with a(p^e) = (4*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)-1))^e(k).

a(n) = A165825(n) * A003958(n) = 4^bigomega(n) * A003958(n) = 4^A001222(n) * A003958(n).

A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 18, 25, 30, 36, 30, 54, 36, 54, 72, 66, 48, 90, 54, 108, 108, 90, 66, 150, 108, 108, 134, 162, 84, 216, 90, 168, 180, 144, 216, 270, 108, 162, 216, 300, 120, 324, 126, 270, 360, 198, 138, 396, 234, 324, 288, 324, 156, 402, 360, 450, 324

Offset: 1

Sebastian Karlsson, Aug 17 2021

Dirichlet convolution of A000010 with A029935.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A000010, A029935, A166633.

f[p_, e_] := (1/2)*(p - 1)*p^(e - 3)*(e^2*(p - 1)^2 + 3*e*(p^2 - 1) + 2*(p^2 + p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 17 2021 *)

from sympy import divisors as div, totient as phi
def D(f, g, n):
    return sum(f(d)*g(n//d) for d in div(n))
def phi_o_phi(n):
    return D(phi, phi, n)
def a(n):
    return D(phi, phi_o_phi, n)

Dirichlet g.f.: zeta(s - 1)^3 / zeta(s)^3.
Multiplicative with a(p^e) = (1/2)*(p-1)*p^(e-3)*(e^2*(p-1)^2 + 3*e*(p^2-1) + 2*(p^2 + p + 1)).
Sum_{k=1..n} a(k) ~ 27*n^2/Pi^10 * (2*Pi^4*log(n)^2 - 2*Pi^4*log(n)*(1 + 6*log(2) - 72*(1/12 - zeta'(-1)) + 6*log(Pi)) + Pi^4*(1 + 6*gamma*(2*gamma - 1) - 12*sg1) + 864*zeta'(2)^2 - 36*Pi^2*((6*gamma - 1)*zeta'(2) + zeta''(2))), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 24 2022

A166635 Totally multiplicative sequence with a(p) = 5*(p-1) for prime p.

Original entry on oeis.org

1, 5, 10, 25, 20, 50, 30, 125, 100, 100, 50, 250, 60, 150, 200, 625, 80, 500, 90, 500, 300, 250, 110, 1250, 400, 300, 1000, 750, 140, 1000, 150, 3125, 500, 400, 600, 2500, 180, 450, 600, 2500, 200, 1500, 210, 1250, 2000, 550, 230, 6250, 900, 2000

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A003958, A165826, A166633, A166634.

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] :=
DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*5^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
a[p_?PrimeQ] := a[p] = 5*(p-1); a[1] = 1; a[n_] := ({pp, ee} = FactorInteger[n] // Transpose; Times @@ ((a /@ pp)^ee)); Array[a, 50] (* Jean-François Alcover, Feb 02 2018 *)

Multiplicative with a(p^e) = (5*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (5*(p(k)-1))^e(k).

a(n) = A165826(n) * A003958(n) = 5^bigomega(n) * A003958(n) = 5^A001222(n) * A003958(n).

cp's OEIS Frontend

A166634 Totally multiplicative sequence with a(p) = 4*(p-1) for prime p.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A166635 Totally multiplicative sequence with a(p) = 5*(p-1) for prime p.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula