A034383 -id:A034383

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A034382

Number of labeled Abelian groups of order n.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

CROSSREFS

Cf. A000688, A034381, A034383, A058159.

KEYWORD

nonn

AUTHOR

Christian G. Bower

EXTENSIONS

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

STATUS

approved

A058163

Number of labeled groups with a fixed identity.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Stephen A. Silver, Table of n, a(n) for n = 1..255

Index entries for sequences related to groups

FORMULA

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

PROG

(GAP) 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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Christian G. Bower, Nov 15 2000

EXTENSIONS

More terms from Stephen A. Silver, Feb 10 2013

STATUS

approved

A034381

Number of labeled cyclic groups with n elements.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Jianing Song, Table of n, a(n) for n = 1..450

Index entries for sequences related to groups

FORMULA

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

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Christian G. Bower

EXTENSIONS

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

STATUS

approved

A058157

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..36.

Index entries for sequences related to monoids

FORMULA

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

EXAMPLE

Triangle begins:

2, 2;

3, 18, 12;

16, 180, 288, 140;

30, 2640, 6540, 8380, 3020;

...

CROSSREFS

Row sums give A058153.

Column 1: A034383.

Main diagonal is A351731.

Cf. A058137 (isomorphism classes), A058158, A058159 (commutative), A058166.

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Nov 14 2000

EXTENSIONS

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

STATUS

approved

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