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

1,2

Links

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

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)

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 *)

Crossrefs

Cf. A000005, A000012, A001620, A034714, A060640.

Sequence in context: A046103 A146459 A308025 * A041158 A146080 A334640

Adjacent sequences: A360426 A360427 A360428 * A360430 A360431 A360432

Keyword

nonn,mult,easy

Author

R. J. Mathar, Feb 07 2023

Status

approved