login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A014221 a(n+1) = 2^a(n) with a(-1) = 0. 66
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 less than n. - Eric M. Schmidt, Jun 29 2013

LINKS

Table of n, a(n) for n=-1..4.

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), pp. 118-133.

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, 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).

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), #07.1.2.

H. M. Friedman, Long finite sequences, J. Comb. Theory, A 95 (2001), 102-144.

Adam P. Goucher, Von Neumann universe (2013)

D. Gijswijt, Een onvoorstelbaar lang woord [An unimaginably long word]

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 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.

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 16 sets of rank less than 4 are:

01: {}

02: {{}}

03: {{}, {{}}}

04: {{{}}}

05: {{}, {{}}, {{}, {{}}}}

06: {{}, {{}}, {{}, {{}}}, {{{}}}}

07: {{}, {{}}, {{{}}}}

08: {{}, {{}, {{}}}}

09; {{}, {{}, {{}}}, {{{}}}}

10: {{}, {{{}}}}

11: {{{}}, {{}, {{}}}}

12: {{{}}, {{}, {{}}}, {{{}}}}

13: {{{}}, {{{}}}}

14: {{{}, {{}}}}

15: {{{}, {{}}}, {{{}}}}

16: {{{{}}}}

(End)

MATHEMATICA

f[n_]:=2^n; p=0; lst={p}; Do[p=f[p]; AppendTo[lst, p], {n, 6}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 03 2009 *)

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)).

Sequence in context: A001128 A124436 * A249760 A271552 A048872 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 2 13:03 EST 2016. Contains 278678 sequences.