Search: id:A014221
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%I A014221
%S A014221 0,1,2,4,16,65536
%N A014221 a(n+1) = 2^a(n) with a(0) = 0. This is the Ackermann function A_3(n+1) 
               as defined in the Comments line..
%C A014221 Next term has 19729 digits - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 28 2002
%C A014221 Harvey Friedman defines the Ackermann function as follows: A_1(n) = 2n, 
               A_{k+1}(n) = A_k A_k ... A_k(1), where there are n A_k's. A_2(n) 
               = 2^n.
%C A014221 Harvey Friedman's rapidly increasing sequence 3, 11, huge, ... does not 
               fit into the constraints of the OEIS. It is described in the paper 
               "Long finite sequences". The third term is > A_7198(158386), which 
               is incomprehensibly huge. See also the Gijswijt article.
%C A014221 The Goodstein sequence described in the Comments in A056041 grows even 
               faster than Friedman's.
%C A014221 a(n) is the smallest a(n-1)-almost prime for n >= 2; e.g. a(5) = 65536 
               = A069277(1) (smallest (a(4)=16)-almost prime). - Rick L. Shepherd 
               (rshepherd2(AT)hotmail.com), Jan 28 2006
%D A014221 David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons, 
               Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.
%D A014221 W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 
               99 (1928), 118-133.
%D A014221 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 A014221 R. C. Buck, Mathematical induction and recursive definitions, Amer. Math. 
               Monthly, 70 (1963), 128-135.
%H A014221 W. Ackermann, <a href="http://134.76.163.65/servlet/digbib?template=view.html&id=29309&startpage=122&endpage=\
               137&image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image-subpath=/
               1417&image-subpath=1417&pagenumber=122&imageset-id=1417">Zum Hilbertschen 
               Aufbau der reellen Zahlen</a>, Math. Ann. 99 (1928), 118-133.
%H A014221 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).
%H A014221 F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and 
               A. R. Wilks, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. 
               Integer Sequences, Vol. 10 (2007), #07.1.2.
%H A014221 H. M. Friedman, <a href="http://www.math.ohio-state.edu/%7Efriedman/pdf/
               finiteseq10_8_98.pdf">Long finite sequences</a>, J. Comb. Theory, 
               A 95 (2001), 102-144.
%H A014221 D. Gijswijt, <a href="http://www.science.uva.nl/~gijswijt/pythagoras/
               GG/deel4/woorden.ps">Een onvoorstelbaar lang woord</a> [An unimaginably 
               long word]
%H A014221 Robert P. Munafo, <a href="http://www.mrob.com/pub/math/seq-a094358.html">
               Sequence A094358, 2^^N = 1 mod N</a>.
%H A014221 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rank.html">Link to a section of The World of Mathematics.</a>
%H A014221 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AckermannFunction.html">Link to a section of The World of Mathematics.</
               a>
%F A014221 a(n) = A004249(n-1)-1. - Leroy Quet, Jun 10 2009
%t A014221 f[n_]:=2^n; p=0;lst={p};Do[p=f[p];AppendTo[lst,p],{n,6}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Jul 03 2009]
%Y A014221 Cf. A038081, A001695, A046859, A093382, A014222, A081651, A114561.
%Y A014221 Cf. A115658 (a(n) is the smallest square-free a(n-1)-almost prime).
%Y A014221 Cf. A007013.
%Y A014221 Sequence in context: A152690 A001128 A124436 this_sequence A048872 A105510 
               A155951
%Y A014221 Adjacent sequences: A014218 A014219 A014220 this_sequence A014222 A014223 
               A014224
%K A014221 nonn,easy,nice
%O A014221 0,3
%A A014221 N. J. A. Sloane (njas(AT)research.att.com).

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