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A014221
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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..
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51
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OFFSET
| 0,3
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COMMENTS
| Next term has 19729 digits - Benoit Cloitre (benoit7848c(AT)orange.fr), 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.
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.
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 (rshepherd2(AT)hotmail.com), 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. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 15 2010]
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REFERENCES
| David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.
W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133.
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.
R. C. Buck, Mathematical induction and recursive definitions, Amer. Math. Monthly, 70 (1963), 128-135.
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LINKS
| W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen, Math. Ann. 99 (1928), 118-133.
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing (arXiv:math.NT/0611293).
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. 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.
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, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| a(n) = A004249(n-1)-1. - Leroy Quet, Jun 10 2009
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MATHEMATICA
| 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]
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CROSSREFS
| Cf. A038081, A001695, A046859, A093382, A014222, A081651, A114561.
Cf. A115658 (a(n) is the smallest squarefree a(n-1)-almost prime).
Cf. A007013.
Sequence in context: A194457 A001128 A124436 * A048872 A105510 A155951
Adjacent sequences: A014218 A014219 A014220 * A014222 A014223 A014224
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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