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A121265 Descending dungeons: a(10)=10; for n>10, a(n) = a(n-1) read as if it were written in base n.

%I A121265 #26 May 11 2024 21:55:51
%S A121265 10,11,13,16,20,30,48,76,132,420,1640,11991,249459,14103793,
%T A121265 5358891675,19563802363305,3359230167951561129,
%U A121265 181335944930584275675841374,54416647690014492928933662292768871352,6605721238793689879501639879905020611382966457124120828,360539645288616164606228883801608423987740093330992456820074646988075733781927268
%N A121265 Descending dungeons: a(10)=10; for n>10, a(n) = a(n-1) read as if it were written in base n.
%C A121265 Using N_b to denote "N read in base b", the sequence is given by
%C A121265 ......10....10.....10.....10.......etc.
%C A121265 ..............11.....11.....11.........
%C A121265 .......................12.....12.......
%C A121265 ................................13.....
%C A121265 where the subscripts are evaluated from the top downwards.
%C A121265 More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".
%C A121265 A "dungeon" of numbers.
%D A121265 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 A121265 N. J. A. Sloane, <a href="/A121265/b121265.txt">Table of n, a(n) for n = 10..35</a>
%H A121265 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 A121265 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 A121265 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 A121265 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
%e A121265 From _Jianing Song_, May 22 2021: (Start)
%e A121265 a(10) = 10;
%e A121265 a(11) = 10_11 = 11;
%e A121265 a(12) = 11_12 = 13;
%e A121265 a(13) = 13_13 = 16;
%e A121265 a(14) = 16_14 = 20;
%e A121265 a(15) = 20_15 = 30;
%e A121265 a(16) = 30_16 = 48;
%e A121265 ... (End)
%p A121265 M:=35; a:=list(10..M): a[10]:=10: lprint(10,a[10]); for n from 11 to M do t1:=convert(a[n-1],base,10); a[n]:=add(t1[i]*n^(i-1),i=1..nops(t1)); lprint(n,a[n]); od:
%t A121265 nxt[{n_,a_}]:={n+1,FromDigits[IntegerDigits[a],n+1]}; Transpose[ NestList[ nxt,{10,10},20]][[2]] (* _Harvey P. Dale_, Jul 13 2014 *)
%o A121265 (PARI) a(n) = {my(x=10); for (b=11, n, x = fromdigits(digits(x, 10), b);); x;} \\ _Michel Marcus_, May 26 2019
%Y A121265 Cf. A121263, A121295, A121296, A127744, A122734.
%K A121265 nonn,base,nice
%O A121265 10,1
%A A121265 _N. J. A. Sloane_, Aug 23 2006