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 1-4 of 4 results.

A344302 Number of cyclic subgroups of the group (C_n)^6, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 64, 365, 2080, 3907, 23360, 19609, 66592, 88817, 250048, 177157, 759200, 402235, 1254976, 1426055, 2130976, 1508599, 5684288, 2613661, 8126560, 7157285, 11338048, 6728905, 24306080, 12210157, 25743040, 21582653, 40786720, 21243691, 91267520, 29583457
Offset: 1

Views

Author

Seiichi Manyama, May 14 2021

Keywords

Comments

Inverse Moebius transform of A160895.

Links

Crossrefs

Cf. A000010, A013664, A060648, A064969, A160895, A280184, A344219, A344303, A344304, A344305, A344306.

Programs

  • Mathematica
    f[p_, e_] := 1 + ((p^6 - 1)/(p - 1))*((p^(5*e) - 1)/(p^5 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a160895(n) = sumdiv(n, d, moebius(n/d)*d^6)/eulerphi(n);
a(n) = sumdiv(n, d, a160895(d));

Formula

a(n) = Sum_{x_1|n, x_2|n, ..., x_6|n} phi(x_1)*phi(x_2)* ... *phi(x_6)/phi(lcm(x_1, x_2, ..., x_6)).
If p is prime, a(p) = 1 + (p^6 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^6 - 1)/(p - 1))*((p^(5*e) - 1)/(p^5 - 1)).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} ((1-1/p^5)/(p^2*(1-1/p))) = 0.32592074105... . (End)

A344303 Number of cyclic subgroups of the group (C_n)^7, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 128, 1094, 8256, 19532, 140032, 137258, 528448, 797891, 2500096, 1948718, 9032064, 5229044, 17569024, 21368008, 33820736, 25646168, 102130048, 49659542, 161256192, 150160252, 249435904, 154764794, 578122112, 305191407, 669317632, 581662904, 1133202048
Offset: 1

Views

Author

Seiichi Manyama, May 14 2021

Keywords

Comments

Inverse Moebius transform of A160897.

Links

Crossrefs

Cf. A000010, A013665, A060648, A064969, A280184, A344219, A344302, A344304, A344305, A344306.

Programs

  • Mathematica
    f[p_, e_] := 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a160897(n) = sumdiv(n, d, moebius(n/d)*d^7)/eulerphi(n);
a(n) = sumdiv(n, d, a160897(d));

Formula

a(n) = Sum_{x_1|n, x_2|n, ..., x_7|n} phi(x_1)*phi(x_2)* ... *phi(x_7)/phi(lcm(x_1, x_2, ..., x_7)).
If p is prime, a(p) = 1 + (p^7 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)).
Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)/7) * Product_{p prime} ((1-1/p^6)/(p^2*(1-1/p))) = 0.2784611791... . (End)

A344304 Number of cyclic subgroups of the group (C_n)^8, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 256, 3281, 32896, 97657, 839936, 960801, 4210816, 7176641, 25000192, 21435889, 107931776, 67977561, 245965056, 320412617, 538984576, 435984841, 1837220096, 943531281, 3212524672, 3152388081, 5487587584, 3559590241, 13815687296, 7629472657, 17402255616
Offset: 1

Views

Author

Seiichi Manyama, May 14 2021

Keywords

Comments

Inverse Moebius transform of A160908.

Links

Crossrefs

Cf. A000010, A013666, A060648, A064969, A160908, A280184, A344219, A344302, A344303, A344305, A344306.

Programs

  • Mathematica
    f[p_, e_] := 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a160908(n) = sumdiv(n, d, moebius(n/d)*d^8)/eulerphi(n);
a(n) = sumdiv(n, d, a160908(d));

Formula

a(n) = Sum_{x_1|n, x_2|n, ..., x_8|n} phi(x_1)*phi(x_2)* ... *phi(x_8)/phi(lcm(x_1, x_2, ..., x_8)).
If p is prime, a(p) = 1 + (p^8 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)).
Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)/8) * Product_{p prime} ((1-1/p^7)/(p^2*(1-1/p))) = 0.2432888374... . (End)

A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 1024, 29525, 524800, 2441407, 30233600, 47079209, 268698112, 581150417, 2500000768, 2593742461, 15494720000, 11488207655, 48209110016, 72082541675, 137573433856, 125999618779, 595098027008, 340614792101, 1281250393600
Offset: 1

Views

Author

Seiichi Manyama, May 14 2021

Keywords

Comments

Inverse Moebius transform of A160957.

Links

Crossrefs

Cf. A000010, A013668, A160957, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344305.

Programs

  • Mathematica
    f[p_, e_] := 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
  • PARI
    a160957(n) = sumdiv(n, d, moebius(n/d)*d^10)/eulerphi(n);
a(n) = sumdiv(n, d, a160957(d));

Formula

a(n) = Sum_{x_1|n, x_2|n, ..., x_10|n} phi(x_1)*phi(x_2)* ... *phi(x_10)/phi(lcm(x_1, x_2, ..., x_10)).
If p is prime, a(p) = 1 + (p^10 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} ((1-1/p^9)/(p^2*(1-1/p))) = 0.1944248708... . (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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