10,1

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.

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.NT/0611293).

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)).

a(13) = 13_(12_(11_10)) = 13_(12_11) = 13_13 = 16.

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;

Cf. A121263, A121265, A121296.

Sequence in context: A093679 A106439 A121263 this_sequence A121296 A121265 A045986

Adjacent sequences: A121292 A121293 A121294 this_sequence A121296 A121297 A121298

nonn,base

David Applegate (david(AT)research.att.com) and njas, Aug 25 2006