cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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.

Original entry on oeis.org

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

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Links

Crossrefs

Cf. A067692, A068021, A068022, A068023, A068024, A068025, A068026, A068027.
Cf. A000203, A001157, A001158.

Programs

  • 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 *)
  • 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

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)

A068022 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=5.

Original entry on oeis.org

1, 63, 364, 2667, 3906, 26964, 19608, 97155, 99463, 271278, 177156, 1228836, 402234, 1324008, 1520784, 3309747, 1508598, 7746453, 2613660, 12021702, 7487424, 11661372, 6728904, 46371780, 12714681, 26297334, 25095280, 57926792
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020, A068021, A068023-A068027, A000203, A001157-A001160.

Programs

  • Mathematica
    CIP5 = CycleIndexPolynomial[SymmetricGroup[5], Array[x, 5]]; a[n_] := CIP5 /. x[k_] -> DivisorSigma[k, n]; Array[a, 28] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/5!*(sigma[1](n)^5 + 10*sigma[1](n)^3*sigma[2](n) + 20*sigma[1](n)^2*sigma[3](n) + 15*sigma[1](n)*sigma[2](n)^2 + 30*sigma[1](n)*sigma[4](n) + 20*sigma[2](n)*sigma[3](n) + 24*sigma[5](n)).

A068023 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=6.

Original entry on oeis.org

1, 127, 1093, 10795, 19531, 164809, 137257, 788035, 896260, 2745247, 1948717, 15172249, 5229043, 18728221, 22858948, 53743987, 25646167, 142560946, 49659541, 244930015, 157475284, 258931921, 154764793, 1151073625, 317886556
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068022, A068024-A068027, A000203, A001157-A001160, A013954.

Programs

  • Mathematica
    CIP6 = CycleIndexPolynomial[SymmetricGroup[6], Array[x, 6]]; a[n_] := CIP6 /. x[k_] -> DivisorSigma[k, n]; Array[a, 25] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/6!*(sigma[1](n)^6 + 15*sigma[1](n)^4*sigma[2](n) + 40*sigma[1](n)^3*sigma[3](n) + 45*sigma[1](n)^2*sigma[2](n)^2 + 90*sigma[1](n)^2*sigma[4](n) + 120*sigma[1](n)*sigma[2](n)*sigma[3](n) + 15*sigma[2](n)^3 + 144*sigma[1](n)*sigma[5](n) + 90*sigma[2](n)*sigma[4](n) + 40*sigma[3](n)^2 + 120*sigma[6](n)).
Agrees with A038994 at n = 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23... - Ralf Stephan, Mar 09 2004

A068024 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=7.

Original entry on oeis.org

1, 255, 3280, 43435, 97656, 998184, 960800, 6347715, 8069620, 27615060, 21435888, 184770040, 67977560, 263540112, 343123440, 866251507, 435984840, 2595218340, 943531280, 4944199260, 3308659904, 5722701624, 3559590240
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068023, A068025-A068027, A000203, A001157-A001160, A013954, A013955.

Programs

  • Mathematica
    CIP7 = CycleIndexPolynomial[SymmetricGroup[7], Array[x, 7]]; a[n_] := CIP7 /. x[k_] -> DivisorSigma[k, n]; Array[a, 23] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/7!*(sigma[1](n)^7 + 21*sigma[1](n)^5*sigma[2](n) + 70*sigma[1](n)^4*sigma[3](n) + 105*sigma[1](n)^3*sigma[2](n)^2 + 210*sigma[1](n)^3*sigma[4](n) + 420*sigma[1](n)^2*sigma[2](n)*sigma[3](n) + 105*sigma[1](n)*sigma[2](n)^3 + 504*sigma[1](n)^2*sigma[5](n) + 630*sigma[1](n)*sigma[2](n)*sigma[4](n) + 280*sigma[1](n)*sigma[3](n)^2 + 210*sigma[2](n)^2*sigma[3](n) + 840*sigma[1](n)*sigma[6](n) + 504*sigma[2](n)*sigma[5](n) + 420*sigma[3](n)*sigma[4](n) + 720*sigma[7](n)).

A068025 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=8.

Original entry on oeis.org

1, 511, 9841, 174251, 488281, 6017605, 6725601, 50955971, 72636421, 276964061, 235794769, 2234070293, 883708281, 3698977205, 5148057541, 13910980083, 7411742281, 46982039533, 17927094321, 99343345101, 69493620405
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068024, A068026-A068027, A000203, A001157-A001160, A013954-A013956.

Programs

  • Mathematica
    CIP8 = CycleIndexPolynomial[SymmetricGroup[8], Array[x, 8]]; a[n_] := CIP8 /. x[k_] -> DivisorSigma[k, n]; Array[a, 21] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/8!*(sigma[1](n)^8 + 28*sigma[1](n)^6*sigma[2](n) + 112*sigma[1](n)^5*sigma[3](n) + 210*sigma[1](n)^4*sigma[2](n)^2 + 420*sigma[1](n)^4*sigma[4](n) + 1120*sigma[1](n)^3*sigma[2](n)*sigma[3](n) + 420*sigma[1](n)^2*sigma[2](n)^3 + 1344*sigma[1](n)^3*sigma[5](n) + 2520*sigma[1](n)^2*sigma[2](n)*sigma[4](n) + 1120*sigma[1](n)^2*sigma[3](n)^2 + 1680*sigma[1](n)*sigma[2](n)^2*sigma[3](n) + 105*sigma[2](n)^4 + 3360*sigma[1](n)^2*sigma[6](n) + 4032*sigma[1](n)*sigma[2](n)*sigma[5](n) + 3360*sigma[1](n)*sigma[3](n)*sigma[4](n) + 1260*sigma[2](n)^2*sigma[4](n) + 1120*sigma[2](n)*sigma[3](n)^2 + 5760*sigma[7](n)*sigma[1](n) + 3360*sigma[2](n)*sigma[6](n) + 2688*sigma[3](n)*sigma[5](n) + 1260*sigma[4](n)^2 + 5040*sigma[8](n)).

A068026 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=9.

Original entry on oeis.org

1, 1023, 29524, 698027, 2441406, 36192156, 47079208, 408345795, 653757313, 2773708938, 2593742460, 26912354924, 11488207654, 51851591352, 77226922344, 222984027123, 125999618778, 848125888467, 340614792100, 1991478050562
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020-A068025, A068027, A000203, A001157-A001160, A013954-A013957.

Programs

  • Mathematica
    CIP9 = CycleIndexPolynomial[SymmetricGroup[9], Array[x, 9]]; a[n_] := CIP9 /. x[k_] -> DivisorSigma[k, n]; Array[a, 20] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/9!*(sigma[1](n)^9 + 36*sigma[1](n)^7*sigma[2](n) + 168*sigma[1](n)^6*sigma[3](n) + 378*sigma[1](n)^5*sigma[2](n)^2 + 756*sigma[1](n)^5*sigma[4](n) + 2520*sigma[1](n)^4*sigma[2](n)*sigma[3](n) +
+ 1260*sigma[1](n)^3*sigma[2](n)^3 + 3024*sigma[1](n)^4*sigma[5](n) + 7560*sigma[1](n)^3*sigma[2](n)*sigma[4](n) + 3360*sigma[1](n)^3*sigma[3](n)^2 + 7560*sigma[1](n)^2*sigma[2](n)^2*sigma[3](n) +
+ 945*sigma[1](n)*sigma[2](n)^4 + 10080*sigma[1](n)^3*sigma[6](n) + 18144*sigma[1](n)^2*sigma[2](n)*sigma[5](n) + 15120*sigma[1](n)^2*sigma[3](n)*sigma[4](n) + 11340*sigma[1](n)*sigma[2](n)^2*sigma[4](n) + 10080*sigma[1](n)*sigma[2](n)*sigma[3](n)^2 + 2520*sigma[2](n)^3*sigma[3](n) + 25920*sigma[7](n)*sigma[1](n)^2 +
+ 30240*sigma[1](n)*sigma[2](n)*sigma[6](n) + 24192*sigma[1](n)*sigma[3](n)*sigma[5](n) + 11340*sigma[1](n)*sigma[4](n)^2 + 9072*sigma[2](n)^2*sigma[5](n) + 15120*sigma[2](n)*sigma[3](n)*sigma[4](n) + 2240*sigma[3](n)^3 + 25920*sigma[7](n)*sigma[2](n) + 45360*sigma[8](n)*sigma[1](n) + 20160*sigma[3](n)*sigma[6](n) + 18144*sigma[4](n)*sigma[5](n) + 40320*sigma[9](n)).

A068021 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=4.

Original entry on oeis.org

1, 31, 121, 651, 781, 4333, 2801, 11811, 11011, 26481, 16105, 96957, 30941, 92613, 100771, 200787, 88741, 412087, 137561, 579201, 354923, 520221, 292561, 1812477, 508431, 993153, 925771, 2003477, 732541, 3996003, 954305, 3309747, 2006851
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2002

Keywords

Crossrefs

Cf. A067692, A068020, A068022-A068027, A000203, A001157, A001158, A001159.

Programs

  • Mathematica
    CIP4 = CycleIndexPolynomial[SymmetricGroup[4], Array[x, 4]]; a[n_] := CIP4 /. x[k_] -> DivisorSigma[k, n]; Array[a, 33] (* Jean-François Alcover, Nov 04 2016 *)

Formula

1/4!*(sigma[1](n)^4 + 6*sigma[1](n)^2*sigma[2](n) + 8*sigma[1](n)*sigma[3](n) + 3*sigma[2](n)^2 + 6*sigma[4](n)).
Showing 1-7 of 7 results.

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