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A344683 Dirichlet convolution of the Euler totient function with itself, applied twice. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Dirichlet convolution of A000010 with A029935.
LINKS
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
Sequence in context: A213685 A271449 A261956 * A166633 A310154 A253277
KEYWORD
nonn,mult
AUTHOR
Sebastian Karlsson, Aug 17 2021
STATUS
approved

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Last modified May 28 21:07 EDT 2023. Contains 363028 sequences. (Running on oeis4.)