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 1-6 of 6 results.

A002489 a(n) = n^(n^2), or (n^n)^n.

Original entry on oeis.org

1, 1, 16, 19683, 4294967296, 298023223876953125, 10314424798490535546171949056, 256923577521058878088611477224235621321607, 6277101735386680763835789423207666416102355444464034512896, 196627050475552913618075908526912116283103450944214766927315415537966391196809
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The number of closed binary operations on a set of order n. Labeled groupoids.
The values of "googol" in base N: "10^100" in base 2 is 2^4=16; "10^100" in base 3 is 3^9=19683, etc. This is N^^3 by the "lower-valued" (left-associative) definition of the hyper4 or tetration operator (see Munafo webpage). - Robert Munafo, Jan 25 2010
n^(n^k) = (((n^n)^n)^...)^n, with k+1 n's, k >= 0. - Daniel Forgues, May 18 2013

Examples

			a(3) = 19683 because (3^3)^3 = 3^(3^2) = 19683.

References

  • John S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 6.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

a(n) = A079172(n) + A023814(n) = A079176(n) + A079179(n);
a(n) = A079182(n) + A023813(n) = A079186(n) + A079189(n);
a(n) = A079192(n) + A079195(n) + A079198(n) + A023815(n).
Cf. A002488, A001329, A002488, A023813, A076113, A090588, A000312, A258102.

Programs

Formula

a(n) = [x^(n^2)] 1/(1 - n*x). - Ilya Gutkovskiy, Oct 10 2017
Sum_{n>=1} 1/a(n) = A258102. - Amiram Eldar, Nov 11 2020

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

Original entry on oeis.org

1, 1, 6, 63, 1140, 30730, 1185072, 66363206, 7150843144, 3829117403448
Offset: 0

Views

Author

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Keywords

Links

Crossrefs

Row sums of A058167.
Cf. A001423, A001426 (isomorphism classes), A023813 (commutative only), A023814 (associative only), A027851.
Cf. A002489, A079192, A079195, A079198, A079201, A079210.

Formula

a(n) + A079192(n) + A079195(n) + A079198(n) = A002489(n).
a(n) = Sum_{k>=1} A079201(n,k)*A079210(n,k). - Andrew Howroyd, Jan 26 2022

Extensions

a(8) from Andrew Howroyd, Jan 26 2022
a(9) from Andrew Howroyd, Feb 14 2022

A079195 Number of non-associative commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 2, 666, 1047436, 30517547395, 21936950639192784, 459986536544739894613595, 324518553658426726783148869733112
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Links

Crossrefs

Cf. A023813, A023815, A079192, A079196 (isomorphism classes), A079197, A079198.

Formula

A079192(n) + a(n) + A079198(n) + A023815(n) = A002489(n).
a(n) = Sum_{k>=1} A079197(n,k)*A079210(n,k).
a(n) = A023813(n) - A023815(n). - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 26 2022

A079193 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 4, 3189, 178937854, 2483527282663335, 14325590003288422852078277, 50976900301814584087291456618542388746
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079193(n)+A079196(n)+A079199(n)+A001426(n)=A001329(n).
Each a(n) is equal to the sum of the elements in row n of A079194.

Links

Crossrefs

Cf. A079192, A079194, A079196, A079199, A001426.
a(n) = A001329(n)-A001425(n)-A027851(n)+A001426(n).
a(n)+A079196(n)+A079199(n)+A001426(n)=A001329(n).

Extensions

Edited and extended by Christian G. Bower, Nov 26 2003

A079198 Number of associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 50, 2352, 153002, 15876046, 7676692858
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

a(n) + A079192(n) + A079195(n) + A023815(n) = A002489(n).
Each a(n) is equal to the sum of the products of each element in row n of A079200 and the corresponding element of A079210.
Since this is the number of labeled noncommutative semigroups on an n-set, a(n) = A023814(n)-A023815(n). - Stanislav Sykora, Apr 03 2016

Links

Crossrefs

Cf. A023814, A023815, A079192, A079195, A079199, A079200.

Extensions

a(5)-a(7) added by Stanislav Sykora, Apr 03 2016

A079194 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 2, 0, 8, 66, 3115, 0, 1, 14, 18, 270, 467, 48260, 178888824
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

Examples

			Triangle T(n,k) begins:
  0;
  0;
  2, 2;
  0, 8, 66, 3115;
  0, 1, 14, 18, 270, 467, 48260, 178888824;
  ...

Links

Crossrefs

Row sums give A079193.
Cf. A027423 (row lengths), A079171, A079192, A079197, A079200, A079201, A079210.

Formula

T(n,k) + A079197(n,k) + A079200(n,k) + A079201(n,k) = A079171(n,k).
A079192(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

Extensions

a(0)=0 prepended by Andrew Howroyd, Jan 26 2022
Showing 1-6 of 6 results.

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

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