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 A014221 a(n+1) = 2^a(n) with a(-1) = 0. 90
 0, 1, 2, 4, 16, 65536 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,3 COMMENTS Also a(n) = H_4(2,n) the tetration (repeated exponentiation) of 2 times n. For definition and key links of H_n(x,y) see A054871. Next term has 19729 digits. - Benoit Cloitre, Mar 28 2002 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). 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. The Goodstein sequence described in the Comments in A056041 grows even faster than Friedman's. 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 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 Number of sets of rank no more than n. - Eric M. Schmidt, Jun 29 2013 [Corrected by Jianing Song, Nov 24 2018] Equivalently, number of sets in the Von Neumann universe V_{n+1}. - Charles R Greathouse IV, Aug 22 2022 LINKS Table of n, a(n) for n = -1..4 Wilhelm Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), pp. 118-133. David Applegate, Marc LeBrun and 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, 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). Bojan Bašić, Paul Ellis, Dana C. Ernst, Danijela Popović, and Nándor Sieben, Categories of impartial rulegraphs and gamegraphs, arXiv:2312.00650 [math.CO], 2023. See p. 17. R. C. Buck, Mathematical induction and recursive definitions, Amer. Math. Monthly, 70 (1963), 128-135. F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), Article 07.1.2. H. M. Friedman, Long finite sequences, J. Comb. Theory, A 95 (2001), 102-144. Dion Gijswijt, Een onvoorstelbaar lang woord [An unimaginably long word], from Internet Archive. Adam P. Goucher, Von Neumann universe (2013). Jack W Grahl, Table of n, a(n) for n = -1..5 Robert P. Munafo, Sequence A094358, 2^^N = 1 mod N. Eric Weisstein's World of Mathematics, Rank. Eric Weisstein's World of Mathematics, Ackermann Function.. Index entries for sequences related to Ackermann function. Index entries for sequences related to Gijswijt's sequence. FORMULA a(n) = H_4(2,n) = 2^^n; a(n) = A_3(n) the Ackermann function defined in the Comments; a(-1) = 0, a(0) = 1, a(n) = 2^2^...^2 (n times); a(n) = A004249(n-1) - 1. - Leroy Quet, Jun 10 2009. Sum_{n>=0} 1/a(n) = A356022. - Amiram Eldar, Jul 30 2022 EXAMPLE a(-1)= H_4(2,-1)= 0; a(0) = H_4(2,0) = 1; a(1) = H_4(2,1) = 2; a(2) = H_4(2,2) = 2^2 = 4; a(3) = H_4(2,3) = 2^2^2 = 16; a(4) = H_4(2,4) = 2^2^2^2 = 65536; From Eric M. Schmidt, Jun 30 2013: (Start) The a(3) = 16 sets of rank no more than 3 are: 01: {} 02: {{}} 03: {{}, {{}}} 04: {{{}}} 05: {{}, {{}}, {{}, {{}}}} 06: {{}, {{}}, {{}, {{}}}, {{{}}}} 07: {{}, {{}}, {{{}}}} 08: {{}, {{}, {{}}}} 09; {{}, {{}, {{}}}, {{{}}}} 10: {{}, {{{}}}} 11: {{{}}, {{}, {{}}}} 12: {{{}}, {{}, {{}}}, {{{}}}} 13: {{{}}, {{{}}}} 14: {{{}, {{}}}} 15: {{{}, {{}}}, {{{}}}} 16: {{{{}}}} (End) MATHEMATICA NestList[2^#&, 0, 6] (* Harvey P. Dale, Dec 19 2012 *) CROSSREFS 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. Sequence in context: A280890 A124436 A369378 * A249760 A271552 A105510 Adjacent sequences: A014218 A014219 A014220 * A014222 A014223 A014224 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane, Jun 14 1998 EXTENSIONS Revision with hyperoperator notation by Natan Arie Consigli Jan 18 2016 STATUS approved

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Last modified April 22 15:37 EDT 2024. Contains 371905 sequences. (Running on oeis4.)