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 N. J. A. 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
KEYWORD
nonn,base
AUTHOR
David Applegate and N. J. A. Sloane, Aug 25 2006
STATUS
approved