

A014221


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


90




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.
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


LINKS

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).
Eric Weisstein's World of Mathematics, Rank.


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);


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;
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



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.


KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



