A360428 Inverse Mobius transformation of A338164.

1, 7, 17, 40, 49, 119, 97, 208, 225, 343, 241, 680, 337, 679, 833, 1024, 577, 1575, 721, 1960, 1649, 1687, 1057, 3536, 1825, 2359, 2673, 3880, 1681, 5831, 1921, 4864, 4097, 4039, 4753, 9000, 2737, 5047, 5729, 10192, 3361, 11543, 3697, 9640, 11025, 7399, 4417, 17408, 7105, 12775

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} A008683(n/d)*A000005(d)*d^2.

Dirichlet convolution of A034714 and A008683.

Dirichlet g.f.: zeta^2(s-2)/zeta(s).

From Amiram Eldar, Feb 09 2023: (Start)

Multiplicative with a(p^e) = (e + 1 - e/p^2)*p^(2*e).

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1/3 - zeta'(3)/zeta(3)) * n^3 / (3*zeta(3)), where gamma is Euler's constant (A001620). (End)

From Peter Bala, Jan 16 2024: (Start)

a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^2. Cf. A069097.

a(n) = Sum_{d divides n} d^2 * J_2(n/d), where J_2(n) = A007434(n). (End)

Maple

A360428 := proc(n)
    add(numtheory[mobius](n/d)*numtheory[tau](d)*d^2, d=numtheory[divisors](n)) ;
end proc:

Mathematica

f[p_, e_] := (e + 1 - e/p^2)*p^(2*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

Crossrefs

Cf. A000005, A001620, A007434, A008683, A018804, A069097, A338164.

Sequence in context: A058273 A058274 A322597 * A193214 A184862 A194772

Adjacent sequences: A360425 A360426 A360427 * A360429 A360430 A360431

Keyword

nonn,mult,easy

Author

R. J. Mathar, Feb 07 2023

Status

approved