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%I #94 Jan 17 2024 12:13:17
%S 1,1,2,9,262144
%N a(0)=1, a(n+1) = (n+1)^a(n).
%C An "exponential factorial".
%C Might also be called the "expofactorial" of n. - Walter Arrighetti (walter.arrighetti(AT)fastwebnet.it), Jan 16 2006
%C By Liouville's theorem, the exponential factorial constant A080219 = Sum_{n>=1} 1/a(n) is a Liouville number and therefore is transcendental. - _Jonathan Sondow_, Jun 17 2014
%D David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.
%D Underwood Dudley, "Mathematical Cranks", MAA 1992, p. 338.
%D F. Luca, D. Marques, Perfect powers in the summatory function of the power tower, J. Theor. Nombr. Bordeaux 22 (3) (2010) 703, doi:10.5802/jtnb.740
%H David Applegate, Marc LeBrun, N. J. A. Sloane, <a href="http://www.jstor.org/stable/40391135">Descending Dungeons, Problem 11286</a>, Amer. Math. Monthly, 116 (2009) 466-467.
%H David Applegate, Marc LeBrun and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0611293">Descending Dungeons and Iterated Base-Changing</a>, arXiv:math.NT/0611293, 2006-2007.
%H Walter Arrighetti, <a href="http://www.die.uniroma1.it/strutture/labcem/">LabCEM</a>, Department of Electronic Engineering, Univ. degli Studi di Roma "La Sapienza".
%H Walter Arrighetti, <a href="http://www.escaped.it/doublevision">Double Vision</a> [Broken link]
%H Vladimir Orlovsky, <a href="http://nrich.maths.org/askedNRICH/edited/75.html">Very Big Number</a>, Feb 19 1999
%H J. Sondow, <a href="http://mathworld.wolfram.com/ExponentialFactorial.html">MathWorld: Exponential Factorial</a>
%H J. Sondow, <a href="http://arXiv.org/abs/math.NT/0406300">Irrationality measures, irrationality bases, and a theorem of Jarnik</a>, arXiv:math/0406300 [math.NT], 2004; see L_4 in Example 4.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Exponential_factorial">Exponential factorial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Liouville_number">Liouville number</a>
%e a(4) = 4^9 = 262144.
%e a(5) = 5^262144 has 183231 decimal digits. - _Rick L. Shepherd_, Feb 15 2002
%e a(5) = ~6.2060698786608744707483205572846793 * 10^183230. - _Robert G. Wilson v_, Oct 24 2015
%e a(6) = 6^(5^262144) has 4.829261036048226... * 10^183230 decimal digits. - _Jack Braxton_, Feb 17 2023
%p a:= proc(n) option remember;
%p `if`(n=0, 1, n^a(n-1))
%p end:
%p seq(a(n), n=0..4); # _Alois P. Heinz_, Jan 17 2024
%t Expofactorial[0] := 1; Expofactorial[n_Integer] := n^Expofactorial[n - 1]; Table[Expofactorial[n], {n, 0, 4}] (* Walter Arrighetti, Jan 24 2006 *)
%t nxt[{n_,a_}]:={n+1,(n+2)^a}; Transpose[NestList[nxt,{0,1},4]][[2]] (* _Harvey P. Dale_, May 26 2013 *)
%o (PARI) a(n)=if(n>1,n^a(n-1),1) \\ _Charles R Greathouse IV_, Sep 13 2013
%Y Cf. A000142, A080219, A140319.
%Y Cf. A132859 (essentially the same).
%K nonn
%O 0,3
%A Marcel Jackson (Marcel.Jackson(AT)utas.edu.au)
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