

A014221


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


81




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_(k1)(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(n1)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(n1) + 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. 118133.
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466467.
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated BaseChanging, 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. 393402. (arXiv:math.NT/0611293).
R. C. Buck, Mathematical induction and recursive definitions, Amer. Math. Monthly, 70 (1963), 128135.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A SlowGrowing 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), 102144.
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(n1)  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(n1)almost prime), A007013, A266198 (a(n) = H_5(2,n)), A356022.
Sequence in context: A001128 A280890 A124436 * 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



