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%I M5030 N2170 #84 Oct 28 2023 11:42:59
%S 1,1,16,19683,4294967296,298023223876953125,
%T 10314424798490535546171949056,
%U 256923577521058878088611477224235621321607,6277101735386680763835789423207666416102355444464034512896,196627050475552913618075908526912116283103450944214766927315415537966391196809
%N a(n) = n^(n^2), or (n^n)^n.
%C The number of closed binary operations on a set of order n. Labeled groupoids.
%C 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 n^(n^k) = (((n^n)^n)^...)^n, with k+1 n's, k >= 0. - _Daniel Forgues_, May 18 2013
%D John S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 6.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Michael Lee, <a href="/A002489/b002489.txt">Table of n, a(n) for n = 0..26</a> (first 16 terms from Vincenzo Librandi)
%H Robert Munafo, <a href="http://mrob.com/pub/math/hyper4.html">Hyper4 Iterated Exponential Function</a> [From _Robert Munafo_, Jan 25 2010]
%H Eric Postpischil, <a href="http://groups.google.com/groups?&hl=en&lr=&ie=UTF-8&selm=11802%40shlump.nac.dec.com&rnum=2">Posting to sci.math newsgroup, May 21 1990</a>.
%H 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://www.digizeitschriften.de/dms/img/?PID=PPN378850199_0003%7Clog8">alternative link</a>.
%H <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</a>
%F a(n) = [x^(n^2)] 1/(1 - n*x). - _Ilya Gutkovskiy_, Oct 10 2017
%F Sum_{n>=1} 1/a(n) = A258102. - _Amiram Eldar_, Nov 11 2020
%e a(3) = 19683 because (3^3)^3 = 3^(3^2) = 19683.
%t Join[{1},Table[n^n^2,{n,10}]] (* _Harvey P. Dale_, Sep 06 2011 *)
%o (Magma) [n^(n^2): n in [0..10]]; // _Vincenzo Librandi_, May 13 2011
%o (PARI) a(n)=n^(n^2) \\ _Charles R Greathouse IV_, Nov 20 2012
%Y a(n) = A079172(n) + A023814(n) = A079176(n) + A079179(n);
%Y a(n) = A079182(n) + A023813(n) = A079186(n) + A079189(n);
%Y a(n) = A079192(n) + A079195(n) + A079198(n) + A023815(n).
%Y Cf. A002488, A001329, A002488, A023813, A076113, A090588, A000312, A258102.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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