A067817 a(n) = Sum_{r|n, s|n, t|n, r

0, 0, 0, 8, 0, 72, 0, 120, 27, 180, 0, 1400, 0, 336, 360, 1240, 0, 3285, 0, 3948, 672, 792, 0, 15960, 125, 1092, 1080, 8240, 0, 25992, 0, 11160, 1584, 1836, 1680, 57065, 0, 2280, 2184, 46620, 0, 56352, 0, 23592, 18612, 3312, 0, 150040, 343, 29955, 3672

Offset: 1

Author

Vladeta Jovovic, Feb 08 2002

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A067692, A068020, A000203, A001157, A001158.

Column k=3 of A224381.

Programs

Maple

a:= n-> coeff(expand(mul(1+d*x, d=numtheory[divisors](n))), x, 3):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 18 2023

Mathematica

a[n_] := Module[{d = DivisorSigma[{1, 2, 3}, n]}, (d[[1]]^3 - 3*d[[1]]*d[[2]] + 2*d[[3]]) / 6]; Array[a, 50] (* Amiram Eldar, Jan 03 2025 *)

PARI

a(n) = 1/6*(sigma(n, 1)^3 - 3*sigma(n, 1)*sigma(n, 2) + 2*sigma(n, 3)) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = (1/3!)*(sigma_1(n)^3 - 3*sigma_1(n)*sigma_2(n) + 2*sigma_3(n)).

From Amiram Eldar, Jan 03 2025: (Start)

Dirichlet g.f.: (zeta(s)*zeta(s-3)/6) * (zeta(s-1)*zeta(s-2) * (f(s) - 3/zeta(2*s-3)) + 2), where f(s) = Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).

Sum_{k=1..n} a(k) ~ c * n^4, where c = (7/96) * zeta(3) * zeta(6) * Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) - zeta(2)*zeta(3)*zeta(4)/(8*zeta(5)) + zeta(4)/12 = 0.085094994884972381542... . (End)