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A049384 a(0)=1, a(n+1) = (n+1)^a(n). 12
1, 1, 2, 9, 262144 (list; graph; refs; listen; history; text; internal format)



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


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.


Table of n, a(n) for n=0..4.

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).

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 2004; see L_4 in Example 4.

Wikipedia, Exponential factorial

Wikipedia, Liouville number


a(4) = 4^9 = 262144.

a(5) = 5^262144 has 183231 decimal digits. - Rick L. Shepherd, Feb 15 2002


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


Cf. A000142, A080219, A140319.

Sequence in context: A028581 A030252 * A132859 A103562 A140319 A120314

Adjacent sequences:  A049381 A049382 A049383 * A049385 A049386 A049387




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



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Last modified November 26 19:22 EST 2014. Contains 250103 sequences.