A002489 a(n) = n^(n^2), or (n^n)^n. (Formerly M5030 N2170)

1, 1, 16, 19683, 4294967296, 298023223876953125, 10314424798490535546171949056, 256923577521058878088611477224235621321607, 6277101735386680763835789423207666416102355444464034512896, 196627050475552913618075908526912116283103450944214766927315415537966391196809

Offset

0,3

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

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

Michael Lee, Table of n, a(n) for n = 0..26 (first 16 terms from Vincenzo Librandi)

Robert Munafo, Hyper4 Iterated Exponential Function [From Robert Munafo, Jan 25 2010]

Eric Postpischil, Posting to sci.math newsgroup, May 21 1990.

P. Rossier, Grands nombres, Elemente der Mathematik, Vol. 3 (1948), p. 20; alternative link.

Index entries for sequences related to groupoids

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

Example

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

Mathematica

Join[{1}, Table[n^n^2, {n, 10}]] (* Harvey P. Dale, Sep 06 2011 *)

Prog

(Magma) [n^(n^2): n in [0..10]]; // Vincenzo Librandi, May 13 2011
(PARI) a(n)=n^(n^2) \\ Charles R Greathouse IV, Nov 20 2012

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.

Sequence in context: A098175 A089232 A300615 * A060205 A140597 A017296

Adjacent sequences: A002486 A002487 A002488 * A002490 A002491 A002492

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved