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 1-10 of 13 results. Next

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

A079196 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 1, 117, 43910, 254429575, 30468670168769, 91267244789189717968, 8048575431238519331999349995, 24051927835861852500932966021639447717, 2755731922430783367615449408031031255128360423993
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Row sums of A079197.
Cf. A001329, A001425, A001426, A079193, A079195 (labeled case), A079199.

Formula

A079193(n) + a(n) + A079199(n) + A001426(n) = A001329(n).
a(n) = A001425(n) - A001426(n). - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and a(5)-a(10) added by Andrew Howroyd, Jan 26 2022

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

A079197 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 107, 0, 0, 0, 5, 0, 28, 488, 43389
Offset: 1

Views

Author

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

Keywords

Comments

A079194(n)+A079197(n)+A079200(n)+A079201(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 0,1; 1,4,5,107; 0,0,0,5,0,28,488,43389
A079195(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079196(x).

Links

Crossrefs

Cf. A079194, A079198, A079199, A079200, A079201.

A079230 Number of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 13026, 3529190912
Offset: 1

Views

Author

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

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079202 and the corresponding element of A079210.

Crossrefs

Cf. A079202, A079231, A079232, A079234, A079236, A079195, A079240, A079242, A079244, A063524.

A079232 Number of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 4, 5826, 764303896
Offset: 1

Views

Author

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

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079203 and the corresponding element of A079210.

Crossrefs

Cf. A079203, A079233, A079230, A079234, A079236, A079195, A079240, A079242, A079244, A063524.

A079234 Number of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 48, 313560
Offset: 1

Views

Author

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

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079204 and the corresponding element of A079210.

Crossrefs

Cf. A079204, A079235, A079230, A079232, A079236, A079195, A079240, A079242, A079244, A063524.

A079236 Number of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 4, 108000
Offset: 1

Views

Author

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

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079205 and the corresponding element of A079210.

Crossrefs

Cf. A079205, A079237, A079230, A079232, A079234, A079195, A079240, A079242, A079244, A063524.

A079244 Number of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 6, 63, 1140
Offset: 1

Views

Author

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

Keywords

Comments

A079230(n)+A079232(n)+A079234(n)+A079236(n)+A079195(n)+A079240(n)+A079242(n)+A079244(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the products of each element in row n of A079209 and the corresponding element of A079210.

Links

Crossrefs

Cf. A079209, A079245, A079230, A079232, A079234, A079236, A079195, A079240, A079242, A063524.

A079240 Number of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 0, 48, 2344, 153000, 15875924, 7676692856, 148188196673360
Offset: 0

Views

Author

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

Keywords

Crossrefs

Cf. A079207, A079241 (isomorphism classes), A079230, A079232, A079234, A079236, A079195, A079242, A079244, A063524.

Formula

A079230(n) + A079232(n) + A079234(n) + A079236(n) + A079195(n) + a(n) + A079242(n) + A079244(n) + A063524(n) = A002489(n).
a(n) = Sum_{k>=1} A079207(n,k)*A079210(n,k).
a(n) = A023814(n) - A023815(n) - A079242(n). - Andrew Howroyd, Jan 27 2022

Extensions

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 27 2022
Showing 1-10 of 13 results. Next

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

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