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.

Previous Showing 91-97 of 97 results.

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)).

A343509 a(n) = Sum_{k=1..n} gcd(k, n)^7.

Original entry on oeis.org

1, 129, 2189, 16514, 78129, 282381, 823549, 2113796, 4787349, 10078641, 19487181, 36149146, 62748529, 106237821, 171024381, 270565896, 410338689, 617568021, 893871757, 1290222306, 1802748761, 2513846349, 3404825469, 4627099444, 6103828145, 8094560241
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Comments

In general, for m > 1, if a(n) = Sum_{j=1..n} gcd(j, n)^m, then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1) * zeta(m+1)). - Vaclav Kotesovec, May 20 2021

Links

Crossrefs

Column 7 of A343510.
Cf. A000010, A013954 (sigma_6(n)), A069092, A343521.

Programs

  • Mathematica
    a[n_] := Sum[GCD[k, n]^7, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    a(n) = sum(k=1, n, gcd(k, n)^7);
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*d^7);
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 6));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+120*x^k+1191*x^(2*k)+2416*x^(3*k)+1191*x^(4*k)+120*x^(5*k)+x^(6*k))/(1-x^k)^8))

Formula

a(n) = Sum_{d|n} phi(n/d) * d^7.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_6(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8.
Dirichlet g.f.: zeta(s-1) * zeta(s-7) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 4725*zeta(7)*n^8 / (4*Pi^8). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_7 <= n} gcd(i_1, ..., i_7, n) = Sum_{d divides n} d * J_7(n/d), where the Jordan totient function J_7(n) = A069092(n). - Peter Bala, Jan 29 2024

A055700 Numbers k such that k | sigma_6(k) + phi(k)^6.

Original entry on oeis.org

1, 2, 6, 22, 84, 350, 525, 1150, 1652, 2366, 2996, 6677, 8140, 8371, 9084, 11510, 11825, 27885, 30694, 43000, 44988, 45060, 49585, 172250, 207194, 312312, 335634, 364084, 568575, 887250, 1183000, 1588956, 1799240, 1829256, 1913975, 2455350, 3523800, 3678454
Offset: 1

Views

Author

Robert G. Wilson v, Jun 09 2000

Keywords

Comments

sigma_6(k) is the sum of the 6th powers of the divisors of k (A013954).

Links

Crossrefs

Cf. A000010, A013954.

Programs

  • Mathematica
    Do[If[Mod[DivisorSigma[6, n]+EulerPhi[n]^6, n]==0, Print[n]], {n, 1, 10^5}]
  • PARI
    isok(n) = !((sigma(n, 6) + eulerphi(n)^6) % n); \\ Michel Marcus, Mar 02 2014

Extensions

More terms from Michel Marcus, Mar 02 2014

A272883 Numbers m such that sigma_k(x) = m has no solution for any k > 0.

Original entry on oeis.org

2, 11, 16, 19, 22, 23, 25, 27, 29, 34, 35, 37, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 58, 59, 61, 64, 66, 67, 69, 70, 71, 75, 76, 77, 79, 81, 83, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119, 123, 125
Offset: 1

Views

Author

Jaroslav Krizek, May 08 2016

Keywords

Comments

Recall that sigma_k(n) = Sum_{d|n} d^k.
Sigma_0(n), the number of divisors of n, can be any positive integer and so is ignored in this sequence.
Complement of A211347.
Numbers n such that A271606(n) = 0.

Links

Crossrefs

Cf. Sequences of sigma_k(n) for k=0-24: A000005 (k=0), A000203 (k=1), A001157-A001160 (k=2-5), A013954-A013972 (k=6-24).
Cf. A211347, A271606.

Programs

  • Magma
    [n: n in [1..2^7] | [n] notsubset Set(Sort([DivisorSigma(k,n): n in [1..2^7+1], k in [1..2^7+1] | DivisorSigma(k,n) lt 2^7+1]))];
Previous Showing 91-97 of 97 results.

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