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A121263 Descending dungeons: see Comments lines for definition.

%I A121263 #39 May 11 2024 21:55:51
%S A121263 10,11,13,16,20,25,31,38,46,55,65,87,135,239,463,943,1967,4143,8751,
%T A121263 18479,38959,103471,306223,942127,2932783,9153583,28562479,89028655,
%U A121263 277145647,861652015,2675637295,10173443119,41132125231,168836688943,695134284847
%N A121263 Descending dungeons: see Comments lines for definition.
%C A121263 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
%C A121263 ......10....10.....10.....10.......etc.
%C A121263 ..............11.....11.....11.........
%C A121263 .......................12.....12.......
%C A121263 ................................13.....
%C A121263 where the subscripts are evaluated from the bottom upwards.
%C A121263 More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".
%C A121263 If a number constructed by iterating exponentials is called a "tower", perhaps these numbers should be called "dungeons".
%C A121263 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.
%C A121263 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.
%D A121263 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.
%H A121263 N. J. A. Sloane, <a href="/A121263/b121263.txt">Table of n, a(n) for n = 10..109</a>
%H A121263 David Applegate, Marc LeBrun and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0611293">Descending Dungeons and Iterated Base-Changing</a>, arXiv:math/0611293 [math.NT], 2006-2007.
%H A121263 David Applegate, Marc LeBrun, and N. J. A. Sloane, <a href="https://www.jstor.org/stable/40391135">Descending Dungeons, Problem 11286</a>, Amer. Math. Monthly, 116 (2009) 466-467.
%H A121263 Brady Haran and N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=xNx3JxRhnZE">Dungeon Numbers</a>, Numberphile video (2020). <a href="https://www.youtube.com/watch?v=HFeKdMf01rQ">(extra)</a>
%F A121263 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
%e A121263 For example,
%e A121263 10
%e A121263 ..11
%e A121263 ....12
%e A121263 ......13
%e A121263 ........14
%e A121263 ..........15
%e A121263 ............16
%e A121263 ..............17
%e A121263 ................18
%e A121263 ..................19
%e A121263 ....................20
%e A121263 ......................21
%e A121263 ........................22
%e A121263 ..........................23
%e A121263 is equal to 239.
%p A121263 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_
%p A121263 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_
%o A121263 (Python)
%o A121263 def a(n):
%o A121263   a_of_n = [((10 + int(i))) for i in range(n)]
%o A121263   while len(a_of_n) != 1:
%o A121263     exponent = 0
%o A121263     a_of_n [-2] = list(str(a_of_n [-2]))
%o A121263     for i in range(len(a_of_n [-2])):
%o A121263       a_of_n [-2] [-(i+1)] = int(a_of_n [-2] [-(i+1)])
%o A121263       a_of_n [-2] [-(i+1)] *= ((a_of_n [-1]) ** exponent)
%o A121263       exponent += 1
%o A121263     a_of_n [-2] = sum(a_of_n [-2])
%o A121263     a_of_n = a_of_n [:((len(a_of_n))-1)]
%o A121263   return (a_of_n [0])
%o A121263 # _Noah J. Crandall_, Dec 07 2020
%Y A121263 Cf. A121266, A121264, A121265, A121295, A121296, A121863, A121864.
%Y A121263 Cf. A122734, A127744, A128916, A130287.
%Y A121263 Cf. A122618 (= n_n), A121802 (the 2-adic limit of this sequence).
%Y A121263 Cf. A049384, A124075.
%K A121263 nonn,nice,base
%O A121263 10,1
%A A121263 _Marc LeBrun_, Aug 23 2006