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

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

Author

Sebastian Karlsson, Aug 17 2021

Keywords

• nonn
• mult

Comments

Dirichlet convolution of A000010 with A029935.

Links

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

Crossrefs

Cf. A000010, A029935, A166633.

Programs

Mathematica

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 *)

Python

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)

Formula

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