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A014221 a(n+1) = 2^a(n) with a(-1) = 0. 90

%I #123 Dec 08 2023 12:05:38

%S 0,1,2,4,16,65536

%N a(n+1) = 2^a(n) with a(-1) = 0.

%C Also a(n) = H_4(2,n) the tetration (repeated exponentiation) of 2 times n.

%C For definition and key links of H_n(x,y) see A054871.

%C Next term has 19729 digits. - _Benoit Cloitre_, Mar 28 2002

%C 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, A_3(n) = 2^^n = H_4(2,n) and A_(k-1)(n) = H_k(2,n).

%C 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 greater than A_7198(158386), which is incomprehensibly huge. See also the Gijswijt article.

%C The Goodstein sequence described in the Comments in A056041 grows even faster than Friedman's.

%C 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_, Jan 28 2006

%C a(0) = 0, for n > 1, a(n) = the smallest number m such that number of divisors of m = previous term + 1, i.e., A000005(a(n)) = a(n-1) + 1. - _Jaroslav Krizek_, Aug 15 2010

%C Number of sets of rank no more than n. - _Eric M. Schmidt_, Jun 29 2013 [Corrected by _Jianing Song_, Nov 24 2018]

%C Equivalently, number of sets in the Von Neumann universe V_{n+1}. - _Charles R Greathouse IV_, Aug 22 2022

%H <a href="/A014221/b014221.txt">Table of n, a(n) for n = -1..4</a>

%H Wilhelm Ackermann, <a href="http://eretrandre.org/rb/files/Ackermann1928_126.pdf">Zum Hilbertschen Aufbau der reellen Zahlen</a>, Math. Ann. 99 (1928), pp. 118-133.

%H David Applegate, Marc LeBrun and N. J. A. Sloane, <a href="http://www.jstor.org/stable/40391135">Descending Dungeons, Problem 11286</a>, Amer. Math. Monthly, 116 (2009) 466-467.

%H 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>, 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. (arXiv:math.NT/0611293).

%H Bojan Bašić, Paul Ellis, Dana C. Ernst, Danijela Popović, and Nándor Sieben, <a href="https://arxiv.org/abs/2312.00650">Categories of impartial rulegraphs and gamegraphs</a>, arXiv:2312.00650 [math.CO], 2023. See p. 17.

%H R. C. Buck, <a href="http://www.jstor.org/stable/2312881">Mathematical induction and recursive definitions</a>, Amer. Math. Monthly, 70 (1963), 128-135.

%H F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Sloane/sloane55.html">A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. Integer Sequences, Vol. 10 (2007), Article 07.1.2.

%H H. M. Friedman, <a href="http://dx.doi.org/10.1006/jcta.2000.3154">Long finite sequences</a>, J. Comb. Theory, A 95 (2001), 102-144.

%H Dion Gijswijt, <a href="https://web.archive.org/web/20031030190740/http://staff.science.uva.nl:80/~gijswijt/pythagoras/GG/deel4/woorden.ps">Een onvoorstelbaar lang woord</a> [An unimaginably long word], from Internet Archive.

%H Adam P. Goucher, <a href="http://cp4space.wordpress.com/2013/07/17/von-neumann-universe/">Von Neumann universe</a> (2013).

%H Jack W Grahl, <a href="/A014221/a014221.txt">Table of n, a(n) for n = -1..5</a>

%H Robert P. Munafo, <a href="http://www.mrob.com/pub/math/seq-a094358.html">Sequence A094358, 2^^N = 1 mod N</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rank.html">Rank</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AckermannFunction.html">Ackermann Function.</a>.

%H <a href="/index/Ab#Ackermann">Index entries for sequences related to Ackermann function</a>.

%H <a href="/index/Ge#Gijswijt">Index entries for sequences related to Gijswijt's sequence</a>.

%F a(n) = H_4(2,n) = 2^^n;

%F a(n) = A_3(n) the Ackermann function defined in the Comments;

%F a(-1) = 0, a(0) = 1, a(n) = 2^2^...^2 (n times);

%F a(n) = A004249(n-1) - 1. - _Leroy Quet_, Jun 10 2009.

%F Sum_{n>=0} 1/a(n) = A356022. - _Amiram Eldar_, Jul 30 2022

%e a(-1)= H_4(2,-1)= 0;

%e a(0) = H_4(2,0) = 1;

%e a(1) = H_4(2,1) = 2;

%e a(2) = H_4(2,2) = 2^2 = 4;

%e a(3) = H_4(2,3) = 2^2^2 = 16;

%e a(4) = H_4(2,4) = 2^2^2^2 = 65536;

%e From _Eric M. Schmidt_, Jun 30 2013: (Start)

%e The a(3) = 16 sets of rank no more than 3 are:

%e 01: {}

%e 02: {{}}

%e 03: {{}, {{}}}

%e 04: {{{}}}

%e 05: {{}, {{}}, {{}, {{}}}}

%e 06: {{}, {{}}, {{}, {{}}}, {{{}}}}

%e 07: {{}, {{}}, {{{}}}}

%e 08: {{}, {{}, {{}}}}

%e 09; {{}, {{}, {{}}}, {{{}}}}

%e 10: {{}, {{{}}}}

%e 11: {{{}}, {{}, {{}}}}

%e 12: {{{}}, {{}, {{}}}, {{{}}}}

%e 13: {{{}}, {{{}}}}

%e 14: {{{}, {{}}}}

%e 15: {{{}, {{}}}, {{{}}}}

%e 16: {{{{}}}}

%e (End)

%t NestList[2^#&,0,6] (* _Harvey P. Dale_, Dec 19 2012 *)

%Y Cf. A038081, A001695, A046859, A093382, A014222 (a(n) = H_4(3,n)), A081651, A114561, A115658 (a(n) is the smallest squarefree a(n-1)-almost prime), A007013, A266198 (a(n) = H_5(2,n)), A356022.

%K nonn,easy,nice

%O -1,3

%A _N. J. A. Sloane_, Jun 14 1998

%E Revision with hyperoperator notation by _Natan Arie Consigli_ Jan 18 2016

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)