A034383 -id:A034383 - cp's OEIS frontend

A034382 Number of labeled Abelian groups of order n.

1, 2, 3, 16, 30, 360, 840, 15360, 68040, 907200, 3991680, 159667200, 518918400, 14529715200, 163459296000, 4250979532800, 22230464256000, 1200445069824000, 6758061133824000, 405483668029440000

Offset: 1

Christian G. Bower

nonn

Max Alekseyev, Table of n, a(n) for n = 1..100
Hy Ginsberg, Totally Symmetric Quasigroups of Order 16, arXiv:2211.13204 [math.CO], 2022.
C. J. Hillar and D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR], 2006.
Sugarknri et al., Number of labeled Abelian groups of order n, Mathematics Stack Exchange, 2019.
Index entries for sequences related to groups

Cf. A000688, A034381, A034383, A058159.

a(n) = A058162(n) * n.

a(16) corrected by Max Alekseyev, Sep 12 2019

A058163 Number of labeled groups with a fixed identity.

Original entry on oeis.org

1, 1, 1, 4, 6, 80, 120, 2760, 7560, 108864, 362880, 21621600, 39916800, 1186099200, 10897286400, 647091244800, 1307674368000, 103742166528000, 355687428096000, 32438693442355200, 260668072304640000, 5573557327822848000, 51090942171709440000

Offset: 1

Christian G. Bower, Nov 15 2000

nonn

Stephen A. Silver, Table of n, a(n) for n = 1..255
Index entries for sequences related to groups

Cf. A000688, A034383, A058158, A058162, A058163.

A058163 := function(n) local fn1, sum, k; fn1 := Factorial(n-1); sum := 0; for k in [1 .. NrSmallGroups(n)] do sum := sum + fn1 / Size(AutomorphismGroup(SmallGroup(n,k))); od; return sum; end; # Stephen A. Silver, Feb 10 2013

a(n) = A034383(n)/n.

More terms from Stephen A. Silver, Feb 10 2013

A034381 Number of labeled cyclic groups with n elements.

Original entry on oeis.org

1, 2, 3, 12, 30, 360, 840, 10080, 60480, 907200, 3991680, 119750400, 518918400, 14529715200, 163459296000, 2615348736000, 22230464256000, 1067062284288000, 6758061133824000, 304112751022080000, 4257578514309120000, 112400072777760768000, 1175091669949317120000

Offset: 1

Christian G. Bower

This sequence is strictly increasing, since a(n) = n!/phi(n) > n!/n = (n-1)! >= a(n-1) for n >= 2. - Jianing Song, Mar 02 2024

Jianing Song, Table of n, a(n) for n = 1..450
Index entries for sequences related to groups

Cf. A000010, A000142, A034382, A034383, A058161.

a(n) = n! / eulerphi(n) \\ Jianing Song, Mar 02 2024

a(n) = n!/phi(n).

a(n) = A000142(n)/A000010(n) = n*A058161(n).

a(21) onwards from Jianing Song, Mar 02 2024

A058157 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents.

Original entry on oeis.org

1, 2, 2, 3, 18, 12, 16, 180, 288, 140, 30, 2640, 6540, 8380, 3020, 480, 119610, 238200, 421020, 372360, 100362, 840, 25196052, 13786290, 26803000, 36174600, 22822674, 4768624, 22080, 48687313640, 2254725312, 2358499080, 3849768160, 3859581096, 1826525120, 305498328

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
   1;
   2,    2;
   3,   18,   12;
  16,  180,  288,  140;
  30, 2640, 6540, 8380, 3020;
  ...

Index entries for sequences related to monoids

Row sums give A058153.
Column 1: A034383.
Main diagonal is A351731.
Cf. A058137 (isomorphism classes), A058158, A058159 (commutative), A058166.

T(n,k) = A058158(n,k)*n.

a(30)-a(36) from Andrew Howroyd, Feb 15 2022

cp's OEIS Frontend

A034382 Number of labeled Abelian groups of order n.

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A058163 Number of labeled groups with a fixed identity.

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A034381 Number of labeled cyclic groups with n elements.

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PARI

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A058157 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents.

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