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.

1, 15, 40, 155, 156, 672, 400, 1395, 1210, 2520, 1464, 7280, 2380, 6336, 6600, 11811, 5220, 21030, 7240, 26880, 16672, 22752, 12720, 66960, 20306, 36792, 33880, 67040, 25260, 119592, 30784, 97155, 60144, 80136, 64080, 230966, 52060, 110880, 97384

Offset

1,2

Links

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

Formula

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

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

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 = 0.60106209766277728837... . (End)

Mathematica

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 *)
CIP3 = CycleIndexPolynomial[SymmetricGroup[3], Array[x, 3]]; a[n_] := CIP3 /. x[k_] -> DivisorSigma[k, n]; Array[a, 39] (* Jean-François Alcover, Nov 04 2016 *)

Prog

(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

Crossrefs

Cf. A067692, A068021, A068022, A068023, A068024, A068025, A068026, A068027.

Cf. A000203, A001157, A001158.

Sequence in context: A175926 A379812 A038991 * A131991 A116042 A219378

Adjacent sequences: A068017 A068018 A068019 * A068021 A068022 A068023

Keyword

nonn,easy

Author

Vladeta Jovovic, Feb 08 2002

Status

approved