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

10,1

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

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 Neil Sloane, Dungeon Numbers, Numberphile video (2020). (extra)

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

Example

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)

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;

Prog

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

Crossrefs

Cf. A121263, A121265, A121296.

Sequence in context: A106439 A290745 A121263 * A121296 A121265 A045986

Adjacent sequences: A121292 A121293 A121294 * A121296 A121297 A121298

Keyword

nonn,base

Author

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

Status

approved