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

Author

R. J. Mathar, Feb 07 2023

Keywords

• nonn
• mult
• easy

Links

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

Crossrefs

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

Programs

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 *)

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)