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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A121263 Descending dungeons: see Comments lines for definition. 16
10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 65, 87, 135, 239, 463, 943, 1967, 4143, 8751, 18479, 38959, 103471, 306223, 942127, 2932783, 9153583, 28562479, 89028655, 277145647, 861652015, 2675637295, 10173443119, 41132125231, 168836688943, 695134284847 (list; graph; refs; listen; history; text; internal format)
OFFSET

10,1

COMMENTS

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.

REFERENCES

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.

LINKS

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/0611293 [math.NT], 2006-2007.

David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

FORMULA

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 and N. J. A. Sloane, Aug 25 2006

EXAMPLE

For example,

10

..11

....12

......13

........14

..........15

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

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

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

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

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

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

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

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

is equal to 239.

MAPLE

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

CROSSREFS

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: A175224 A106439 A290745 * A121295 A121296 A121265

Adjacent sequences:  A121260 A121261 A121262 * A121264 A121265 A121266

KEYWORD

nonn,nice,base

AUTHOR

Marc LeBrun, Aug 23 2006

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 October 18 01:01 EDT 2018. Contains 316297 sequences. (Running on oeis4.)