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 1-8 of 8 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

A079177 Number of isomorphism classes of non-anti-associative closed binary operations on a set of order n.

Original entry on oeis.org

0, 8, 3320, 178964172
Offset: 1

Views

Author

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

Keywords

Comments

A079177(n)+A079180(n)=A001329(n).
Each a(n) is equal to the sum of the elements in row n of A079178.

Links

Crossrefs

Cf. A079176, A079178, A079180.

A079178 Number of isomorphism classes of non-anti-associative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 6, 3, 10, 78, 3229, 2, 1, 12, 30, 246, 495, 48427, 178914959
Offset: 1

Views

Author

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

Keywords

Comments

A079178(n)+A079181(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; 2,6; 3,10,78,3229; 2,1,12,30,246,495,48427,178914959
A079176(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 A079177(x).

Links

Crossrefs

Cf. A079176, A079177, A079179.

A079179 Number of anti-associative closed binary operations on a set of order n.

Original entry on oeis.org

1, 0, 2, 52, 421560
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A079176, A079180 (isomorphism classes), A079181, A079210.

Formula

a(n) = A002489(n) - A079176(n).
a(n) = Sum_{k>=1} A079181(n,k)*A079210(n,k).

Extensions

a(0)=1 prepended and a(1) corrected by Kamil Zabielski, Aug 28 2024

A079184 Number of isomorphism classes of non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 4, 2, 2, 8, 70, 3121, 2, 1, 14, 22, 275, 467, 48306, 178888897
Offset: 1

Views

Author

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

Keywords

Comments

A079184(n)+A079185(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; 4,2; 2,8,70,3121; 2,1,14,22,275,467,48306,178888897
A079176(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 n is given by A079177(n).

Links

Crossrefs

Cf. A079182, A079183, A079185.

A079181 Number of isomorphism classes of anti-associative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 8, 0, 0, 2, 0, 29, 0, 383, 17366
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

			First rows:
  1;
  0;
  2,0;
  0,2,0,8;
  0,0,2,0,29,0,383,17366;
  ...

Links

Crossrefs

Cf. A027423 (row lengths), A079176, A079177, A079180 (row sums).

Formula

T(n,k) = A079171(n,k) - A079178(n,k).
A079179(n) = Sum_{k>=1} A079210(n,k) * T(n,k).

Extensions

a(0)=1 prepended and a(1) corrected by Kamil Zabielski, Aug 28 2024

A079185 Number of isomorphism classes of commutative closed binary operations (groupoids) on a set of order n, listed by class size.

Original entry on oeis.org

1, 0, 4, 1, 4, 8, 116, 0, 0, 0, 8, 0, 28, 504, 43428
Offset: 1

Views

Author

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

Keywords

Comments

A079184(n)+A079185(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 1; 0,4; 1,4,8,116; 0,0,0,8,0,28,504,43428
A079176(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 n is given by A079177(n).

Links

Crossrefs

Cf. A001425, A023183, A079184. a(n, A027423(n)) = A030255(n).

A079188 Number of isomorphism classes of non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 4, 1, 4, 44, 2285, 0, 0, 0, 24, 64, 212, 35240, 147088764
Offset: 1

Views

Author

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

Keywords

Comments

Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
A079176(n) is equal to the sum of the products of each element in row n of this sequence and the corresponding element of A079210.
The sum of each row n of this sequence is given by A079177(n).

Examples

			First four rows:
  0;
  0, 4;
  1, 4, 44, 2285;
  0, 0, 0, 24, 64, 212, 35240, 147088764.

Links

Crossrefs

Cf. A027423, A079171, A079186, A079187, A079191.

Formula

a(n) = A079171(n) - A079191(n).
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.