cp's OEIS Frontend

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

%I A002489 M5030 N2170 #86 Aug 11 2025 08:34:07
%S A002489 1,1,16,19683,4294967296,298023223876953125,
%T A002489 10314424798490535546171949056,
%U A002489 256923577521058878088611477224235621321607,6277101735386680763835789423207666416102355444464034512896,196627050475552913618075908526912116283103450944214766927315415537966391196809
%N A002489 a(n) = n^(n^2), or (n^n)^n.
%C A002489 The number of closed binary operations on a set of order n. Labeled groupoids.
%C A002489 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
%C A002489 n^(n^k) = (((n^n)^n)^...)^n, with k+1 n's, k >= 0. - _Daniel Forgues_, May 18 2013
%D A002489 John S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 6.
%D A002489 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002489 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002489 Michael Lee, <a href="/A002489/b002489.txt">Table of n, a(n) for n = 0..26</a> (first 16 terms from Vincenzo Librandi)
%H A002489 Robert Munafo, <a href="http://mrob.com/pub/math/hyper4.html">Hyper4 Iterated Exponential Function</a> [From _Robert Munafo_, Jan 25 2010]
%H A002489 Eric Postpischil, <a href="http://groups.google.com/groups?&amp;hl=en&amp;lr=&amp;ie=UTF-8&amp;selm=11802%40shlump.nac.dec.com&amp;rnum=2">Posting to sci.math newsgroup, May 21 1990</a>.
%H A002489 P. Rossier, <a href="http://retro.seals.ch/digbib/view?pid=elemat-001:1948:3::26">Grands nombres</a>, Elemente der Mathematik, Vol. 3 (1948), p. 20; <a href="https://gdz.sub.uni-goettingen.de/id/PPN378850199_0003">alternative link</a>.
%H A002489 <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</a>
%F A002489 a(n) = [x^(n^2)] 1/(1 - n*x). - _Ilya Gutkovskiy_, Oct 10 2017
%F A002489 Sum_{n>=1} 1/a(n) = A258102. - _Amiram Eldar_, Nov 11 2020
%e A002489 a(3) = 19683 because (3^3)^3 = 3^(3^2) = 19683.
%t A002489 Join[{1},Table[n^n^2,{n,10}]] (* _Harvey P. Dale_, Sep 06 2011 *)
%o A002489 (Magma) [n^(n^2): n in [0..10]]; // _Vincenzo Librandi_, May 13 2011
%o A002489 (PARI) a(n)=n^(n^2) \\ _Charles R Greathouse IV_, Nov 20 2012
%Y A002489 a(n) = A079172(n) + A023814(n) = A079176(n) + A079179(n);
%Y A002489 a(n) = A079182(n) + A023813(n) = A079186(n) + A079189(n);
%Y A002489 a(n) = A079192(n) + A079195(n) + A079198(n) + A023815(n).
%Y A002489 Cf. A002488, A001329, A002488, A023813, A076113, A090588, A000312, A258102.
%K A002489 nonn,easy,nice
%O A002489 0,3
%A A002489 _N. J. A. Sloane_