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A121296 Descending dungeons: like A121295 but read subscripts from top downwards.

%I A121296 #16 May 11 2024 21:55:51
%S A121296 10,11,13,16,20,28,45,73,133,348,4943,22779,537226,11662285,
%T A121296 46524257772,1092759075796059,159271598072111595659,
%U A121296 3317896028408943302861454961,594387514787460257685718548861374076357,91930654519343922607883279072515432244874866615525276
%N A121296 Descending dungeons: like A121295 but read subscripts from top downwards.
%C A121296 A "dungeon" of numbers.
%D A121296 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 A121296 N. J. A. Sloane, <a href="/A121296/b121296.txt">Table of n, a(n) for n = 10..35</a>
%H A121296 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 A121296 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 A121296 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 A121296 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)).
%e A121296 a(13) = ((13_12)_11)_10 = (15_11)_10 = 16_10 = 16.
%p A121296 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 A121296 s2:=[10]; for n from 11 to 35 do t1:=n; for i from 1 to n-10 do t1:=asubb(t1,n-i); od: s2:=[op(s2),t1]; od;
%Y A121296 Cf. A121263, A121265, A121295.
%K A121296 nonn,base
%O A121296 10,1
%A A121296 _David Applegate_ and _N. J. A. Sloane_, Aug 25 2006