

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
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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 baseb 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 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.


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 BaseChanging, arXiv:math/0611293 [math.NT], 20062007.
David Applegate, Marc LeBrun, and N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466467.
Brady Haran and Neil Sloane, Dungeon Numbers, Numberphile video (2020). (extra)


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_{i=1..n} log(9+i)), 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 n1 by 1 to 11 do t1:=convert(i, base, 10); b:=add(t1[j]*b^(j1), 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^(j1), j=1..nops(t1)): end; # asubb(a, b) evaluates a as if it were written in base b # N. J. A. Sloane


PROG

(Python)
def a(n):
a_of_n = [((10 + int(i))) for i in range(n)]
while len(a_of_n) != 1:
exponent = 0
a_of_n [2] = list(str(a_of_n [2]))
for i in range(len(a_of_n [2])):
a_of_n [2] [(i+1)] = int(a_of_n [2] [(i+1)])
a_of_n [2] [(i+1)] *= ((a_of_n [1]) ** exponent)
exponent += 1
a_of_n [2] = sum(a_of_n [2])
a_of_n = a_of_n [:((len(a_of_n))1)]
return (a_of_n [0])
# Noah J. Crandall, Dec 07 2020


CROSSREFS

Cf. A121266, A121264, A121265, A121295, A121296, A121863, A121864.
Cf. A122734, A127744, A128916, A130287.
Cf. A122618 (= n_n), A121802 (the 2adic 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



