A067817 - OEIS

%I #18 Jan 03 2025 09:37:18

%S 0,0,0,8,0,72,0,120,27,180,0,1400,0,336,360,1240,0,3285,0,3948,672,

%T 792,0,15960,125,1092,1080,8240,0,25992,0,11160,1584,1836,1680,57065,

%U 0,2280,2184,46620,0,56352,0,23592,18612,3312,0,150040,343,29955,3672

%N a(n) = Sum_{r|n, s|n, t|n, r<s<t} r*s*t.

%H Alois P. Heinz, <a href="/A067817/b067817.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Amiram Eldar_, Jan 03 2025: (Start)

%F 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)).

%F 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)

%p a:= n-> coeff(expand(mul(1+d*x, d=numtheory[divisors](n))), x, 3):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 18 2023

%t 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 *)

%o (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

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

%Y Column k=3 of A224381.

%K nonn

%O 1,4

%A _Vladeta Jovovic_, Feb 08 2002