10,1

Let "N_b" denote "N read in base b" and let "N" denote "N written in base 10" (as in normal life). The sequence is given by 10, 10_11, 10_(11_12), 10_(11_(12_13)), 10_(11_(12_(13_14))), etc., or in other words

......10....10.....10.....10.......etc.

..............11.....11.....11.........

.......................12.....12.......

................................13.....

where the subscripts are evaluated from the bottom upwards.

More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".

If a number constructed by iterating exponentials is called a "tower", perhaps these numbers should be called "dungeons".

The sequence has steady growth until a(101) but then speeds up - see the extended table. For n <= 100, a(n) grows by less than a factor of 10 each iteration. For n >= 100, a(n)/a(99) at least squares each iteration. After a(1000) it will accelerate again and so on.

This is one of a family of four related sequences: alpha: A121263 (this sequence), beta: A121265, gamma: A121295, delta: A121296. The four main difference sequences are beta - alpha: A122734, beta - gamma: A127744, delta - alpha: A130287 and delta - gamma: A128916. The other two differences are gamma - alpha: A131011 and delta - beta: A131012.

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.

N. J. A. Sloane, Table of n, a(n) for n = 10..109

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing (arXiv:math.NT/0611293).

If a, b >= 10, then a_b is roughly 10^(log(a)log(b)) (all logs are base 10 and "roughly" means it is an upper bound and using floor(log()) gives a lower bound). Equivalently, there exists c > 0 such that for all a, b >= 10, 10^(c log(a)log(b)) <= a_b <= 10^(log(a)log(b)). Thus a_n is roughly 10^product(log(9+i),i=1..n), or equivalently, a_n = 10^10^(n loglog n + O(n)). - David Applegate (david(AT)research.att.com) and N. J. A. Sloane (njas(AT)research.att.com), Aug 25 2006

For example,

10

..11

....12

......13

........14

..........15

............16

..............17

................18

..................19

....................20

......................21

........................22

..........................23

is equal to 239.

M:=100; a:=list(10..M): a[10]:=10: lprint(10, a[10]); for n from 11 to M do b:=n; for i from n-1 by -1 to 11 do t1:=convert(i, base, 10); b:=add(t1[j]*b^(j-1), j=1..nops(t1)): od: a[n]:=b; lprint(n, a[n]); od: (N. J. A. Sloane)

asubb := proc(a, b) local t1; t1:=convert(a, base, 10); add(t1[j]*b^(j-1), j=1..nops(t1)): end; # asubb(a, b) evaluates a as if it were written in base b (N. J. A. Sloane)

Cf. A121266, A121264, A121265, A121295, A121296, A121863, A121864.

Cf. A122734, A127744, A128916, A130287.

Cf. A122618 (= n_n), A121802 (the 2-adic limit of this sequence).

Cf. A049384, A124075.

Sequence in context: A093679 A153194 A106439 this_sequence A121295 A121296 A121265

Adjacent sequences: A121260 A121261 A121262 this_sequence A121264 A121265 A121266

nonn,nice,base

Marc LeBrun (mlb(AT)well.com), Aug 23 2006