A080219 -id:A080219 - cp's OEIS frontend

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

1, 1, 2, 9, 262144

Offset: 0

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

nonn

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

			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

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

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

Cf. A000142, A080219, A140319.

Cf. A132859 (essentially the same).

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

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 *)

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

A167155 Exponential primorial constant Sum_{k>=0} 1/A140319(k).

Original entry on oeis.org

1, 6, 1, 1, 1, 1, 1, 6, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

M. F. Hasler, Nov 03 2009

This is a Liouville number and therefore transcendental.

			1 + 1/2^1 + 1/3^2 + 1/5^9 + 1/7^(5^9)+ ... = 1.6111116231111111111111111111111111111111...
Since 1/9 = 0.11111... and 1/5^9 = 512*10^(-9), the initial 10 digits are 1.611111623.
Since 1/A140319(4) = 1/7^1953125 = 7.7731519...*10^(-1650583), these digits are followed by a string of 1650573 "1"s, then followed by digits 8884263011....

J. Sondow, Exponential Factorial.
Index entries for transcendental numbers

Cf. A080219.

Clear[ep, s]; ep[0] = 1; ep[n_] := Prime[n]^ep[n-1]; s[n_] := s[n] = RealDigits[Sum[1/ep[k], {k, 0, n}], 10, 105] // First; s[n=1]; While[s[n] != s[n-1], n++]; s[n] (* Jean-François Alcover, Feb 13 2013 *)

1+1/2+1/3^2+1/5^9+1/7^5^9. /* The final dot is part of the code! */

cp's OEIS Frontend

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

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