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A068020 a(n) = Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=3.

%I A068020 #23 Jan 03 2025 09:37:11
%S A068020 1,15,40,155,156,672,400,1395,1210,2520,1464,7280,2380,6336,6600,
%T A068020 11811,5220,21030,7240,26880,16672,22752,12720,66960,20306,36792,
%U A068020 33880,67040,25260,119592,30784,97155,60144,80136,64080,230966,52060,110880,97384
%N A068020 a(n) = Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=3.
%H A068020 Amiram Eldar, <a href="/A068020/b068020.txt">Table of n, a(n) for n = 1..10000</a>
%F A068020 a(n) = (1/3!)*(sigma_1(n)^3 + 3*sigma_1(n)*sigma_2(n) + 2*sigma_3(n)).
%F A068020 a(n) = Sum_{r|n, s|n, t|n, r<=s<=t} r*s*t.
%F A068020 From _Amiram Eldar_, Jan 03 2025: (Start)
%F A068020 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 A068020 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 = 0.60106209766277728837... . (End)
%t A068020 a[n_] := 1/3!*(DivisorSigma[1, n]^3 + 3*DivisorSigma[1, n]*DivisorSigma[2, n] + 2*DivisorSigma[3, n]); Table[a[n], {n, 1, 39}] (* _Jean-François Alcover_, Dec 12 2011, after given formula *)
%t A068020 CIP3 = CycleIndexPolynomial[SymmetricGroup[3], Array[x, 3]]; a[n_] := CIP3 /. x[k_] -> DivisorSigma[k, n]; Array[a, 39] (* _Jean-François Alcover_, Nov 04 2016 *)
%o A068020 (PARI) a(n) = my(f = factor(n)); (2*sigma(f, 3) + 3*sigma(f, 1)*sigma(f, 2) + sigma(f)^3) / 6; \\ _Amiram Eldar_, Jan 03 2025
%Y A068020 Cf. A067692, A068021, A068022, A068023, A068024, A068025, A068026, A068027.
%Y A068020 Cf. A000203, A001157, A001158.
%K A068020 nonn,easy
%O A068020 1,2
%A A068020 _Vladeta Jovovic_, Feb 08 2002