OFFSET
0,3
COMMENTS
An "exponential factorial".
Might also be called the "expofactorial" of n. - Walter Arrighetti (walter.arrighetti(AT)fastwebnet.it), Jan 16 2006
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
REFERENCES
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.
Underwood Dudley, "Mathematical Cranks", MAA 1992, p. 338.
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
LINKS
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math.NT/0611293, 2006-2007.
Walter Arrighetti, LabCEM, Department of Electronic Engineering, Univ. degli Studi di Roma "La Sapienza".
Walter Arrighetti, Double Vision [Broken link]
Vladimir Orlovsky, Very Big Number, Feb 19 1999
J. Sondow, MathWorld: Exponential Factorial
J. Sondow, Irrationality measures, irrationality bases, and a theorem of Jarnik, arXiv:math/0406300 [math.NT], 2004; see L_4 in Example 4.
Wikipedia, Exponential factorial
Wikipedia, Liouville number
EXAMPLE
a(4) = 4^9 = 262144.
a(5) = 5^262144 has 183231 decimal digits. - Rick L. Shepherd, Feb 15 2002
a(5) = ~6.2060698786608744707483205572846793 * 10^183230. - Robert G. Wilson v, Oct 24 2015
a(6) = 6^(5^262144) has 4.829261036048226... * 10^183230 decimal digits. - Jack Braxton, Feb 17 2023
MAPLE
a:= proc(n) option remember;
`if`(n=0, 1, n^a(n-1))
end:
seq(a(n), n=0..4); # Alois P. Heinz, Jan 17 2024
MATHEMATICA
Expofactorial[0] := 1; Expofactorial[n_Integer] := n^Expofactorial[n - 1]; Table[Expofactorial[n], {n, 0, 4}] (* Walter Arrighetti, Jan 24 2006 *)
nxt[{n_, a_}]:={n+1, (n+2)^a}; Transpose[NestList[nxt, {0, 1}, 4]][[2]] (* Harvey P. Dale, May 26 2013 *)
PROG
(PARI) a(n)=if(n>1, n^a(n-1), 1) \\ Charles R Greathouse IV, Sep 13 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Marcel Jackson (Marcel.Jackson(AT)utas.edu.au)
STATUS
approved