A121295 Descending dungeons: for definition see Comments lines.

10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 110, 221, 444, 891, 1786, 3577, 7160, 14327, 28662, 57333, 171999, 515998, 1547996, 4643991, 13931977, 41795936, 125387814, 376163449, 1128490355, 3385471074, 13541884296, 54167537185, 216670148742, 866680594971

Offset: 10

Author

David Applegate and N. J. A. Sloane, Aug 25 2006

Keywords

• nonn
• base

Comments

Using N_b to denote "N read in base b", the sequence is

......10....11.....12.....13.......etc.

..............10.....11.....12.........

.......................10.....11.......

................................10.....

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 base-b expansion".

A "dungeon" of numbers.

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

Examples

			a(13) = 13_(12_(11_10)) = 13_(12_11) = 13_13 = 16.
From _Jianing Song_, May 22 2021: (Start)
a(10) = 10;
a(11) = 11_10 = 11;
a(12) = 12_11 = 13;
a(13) = 13_13 = 16;
a(14) = 14_16 = 20;
a(15) = 15_20 = 25;
a(16) = 16_25 = 31;
... (End)
		

References

• 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.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 10..103
• David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
• David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.
• Brady Haran and N. J. A. Sloane, Dungeon Numbers, Numberphile video (2020). (extra)

Crossrefs

Cf. A121263, A121265, A121296.

Programs

Maple

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
s1:=[10]; for n from 11 to 50 do i:=n-10; s1:=[op(s1), asubb(n,s1[i])]; od: s1;

PARI

a(n) = {my(x=10); for (b=11, n, x = fromdigits(digits(b, 10), x);); x;} \\ Michel Marcus, May 26 2019

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(log(9+i),i=1..n), or equivalently, a_n = 10^10^(n loglog n + O(n)).

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