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%I #41 May 23 2021 00:19:36
%S 10,11,13,16,20,25,31,38,46,55,110,221,444,891,1786,3577,7160,14327,
%T 28662,57333,171999,515998,1547996,4643991,13931977,41795936,
%U 125387814,376163449,1128490355,3385471074,13541884296,54167537185,216670148742,866680594971
%N Descending dungeons: for definition see Comments lines.
%C Using N_b to denote "N read in base b", the sequence is
%C ......10....11.....12.....13.......etc.
%C ..............10.....11.....12.........
%C .......................10.....11.......
%C ................................10.....
%C where the subscripts are evaluated from the bottom upwards.
%C More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".
%C A "dungeon" of numbers.
%C a(10) = 10; for n > 10, a(n) = n read as if it were written in base a(n-1). - _Jianing Song_, May 22 2021
%D 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 N. J. A. Sloane, <a href="/A121295/b121295.txt">Table of n, a(n) for n = 10..103</a>
%H 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 David Applegate, Marc LeBrun, 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 Brady Haran and Neil 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 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)).
%F A121295(10) = 10, A121295(n) = Sum_{i=0..m-1} A121295(n-1)^(m-1-i) * d_(m-i), for n >= 11, where n = d_m,...,d_2,d_1 is the decimal expansion of n. - _Christopher Hohl_, Jun 11 2019
%e a(13) = 13_(12_(11_10)) = 13_(12_11) = 13_13 = 16.
%e From _Jianing Song_, May 22 2021: (Start)
%e a(10) = 10;
%e a(11) = 11_10 = 11;
%e a(12) = 12_11 = 13;
%e a(13) = 13_13 = 16;
%e a(14) = 14_16 = 20;
%e a(15) = 15_20 = 25;
%e a(16) = 16_25 = 31;
%e ... (End)
%p 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
%p s1:=[10]; for n from 11 to 50 do i:=n-10; s1:=[op(s1), asubb(n,s1[i])]; od: s1;
%o (PARI) a(n) = {my(x=10); for (b=11, n, x = fromdigits(digits(b, 10), x);); x;} \\ _Michel Marcus_, May 26 2019
%Y Cf. A121263, A121265, A121296.
%K nonn,base
%O 10,1
%A _David Applegate_ and _N. J. A. Sloane_, Aug 25 2006
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