10,1

A "dungeon" of numbers.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

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.

N. J. A. Sloane, Table of n, a(n) for n = 10..35

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 = (15_11)_10 = 16_10 = 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

s2:=[10]; for n from 11 to 35 do t1:=n; for i from 1 to n-10 do t1:=asubb(t1, n-i); od: s2:=[op(s2), t1]; od;

Cf. A121263, A121265, A121295.

Sequence in context: A106439 A121263 A121295 this_sequence A121265 A045986 A125588

Adjacent sequences: A121293 A121294 A121295 this_sequence A121297 A121298 A121299

nonn,base

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