A049384 a(0)=1, a(n+1) = (n+1)^a(n).

1, 1, 2, 9, 262144

Offset: 0

Author

Marcel Jackson (Marcel.Jackson(AT)utas.edu.au)

Keywords

• nonn

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

Examples

			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
		

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

Crossrefs

Cf. A000142, A080219, A140319.

Cf. A132859 (essentially the same).

Programs

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 *)

PARI

a(n)=if(n>1,n^a(n-1),1) \\ Charles R Greathouse IV, Sep 13 2013