A360429 Inverse Mobius transformation of A034714.

1, 9, 19, 57, 51, 171, 99, 313, 262, 459, 243, 1083, 339, 891, 969, 1593, 579, 2358, 723, 2907, 1881, 2187, 1059, 5947, 1926, 3051, 3178, 5643, 1683, 8721, 1923, 7737, 4617, 5211, 5049, 14934, 2739, 6507, 6441, 15963, 3363, 16929, 3699, 13851, 13362, 9531, 4419, 30267, 7302, 17334

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, A000012, A001620, A034714, A060640.

Programs

Maple

A360429 := proc(n)
    add(numtheory[tau](d)*d^2,d=numtheory[divisors](n)) ;
end proc:

Mathematica

f[p_, e_] := ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 09 2023 *)

Formula

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

Dirichlet convolution of A034714 and A000012.

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

From Amiram Eldar, Feb 09 2023: (Start)

Multiplicative with a(p^e) = ((e+1)*p^(2*e+4) - (e+2)*p^(2*e+2) + 1)/(p^2-1)^2.

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