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

1,2

Comments

Dirichlet convolution of A000010 with A029935.

Links

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

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

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 *)

Prog

(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)

Crossrefs

Cf. A000010, A029935, A166633.

Sequence in context: A213685 A271449 A261956 * A166633 A310154 A253277

Adjacent sequences: A344680 A344681 A344682 * A344684 A344685 A344686

Keyword

nonn,mult

Author

Sebastian Karlsson, Aug 17 2021

Status

approved