A386012 a(n) = n^3*tau(n).

1, 16, 54, 192, 250, 864, 686, 2048, 2187, 4000, 2662, 10368, 4394, 10976, 13500, 20480, 9826, 34992, 13718, 48000, 37044, 42592, 24334, 110592, 46875, 70304, 78732, 131712, 48778, 216000, 59582, 196608, 143748, 157216, 171500, 419904, 101306, 219488, 237276, 512000

Offset: 1

Author

R. J. Mathar, Jul 14 2025

Keywords

• nonn
• mult
• easy

Comments

Dirichlet convolution of the cubes A000578 with themselves.

Links

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

Crossrefs

Cf. A000005, A001620, A034714, A038040, A320895 (partial sums), A372928 (Mobius transform).

Programs

Maple

seq( n^3*numtheory[tau](n),n=1..100) ;

Mathematica

a[n_]:=n^3*DivisorSigma[0,n]; Array[a,40] (* Stefano Spezia, Jul 14 2025 *)
nmax = 40; Rest[CoefficientList[Series[Sum[k^3*x^k*(1 + 4*x^k + x^(2*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 03 2025 *)

PARI

a(n) = n^3 * numdiv(n); \\ Amiram Eldar, Jul 15 2025

Formula

a(n) = n*A034714(n) = n^2*A038040(n).

Dirichlet g.f.: zeta^2(s-3).

From Amiram Eldar, Jul 15 2025 (Start)

Multiplicative with a(p^e) = p^(3*e) * (e+1).

Sum_{k=1..n} a(k) ~ (n^4/4) * (log(n) + 2*gamma - 1/4), where gamma is Euler's constant (A001620). (End)

G.f.: Sum_{k>=1} k^3*x^k*(1 + 4*x^k + x^(2*k)) / (1-x^k)^4. - Vaclav Kotesovec, Aug 03 2025