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A121295 Descending dungeons: for definition see Comments lines. 14
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 (list; graph; refs; listen; history; text; internal format)
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.

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.

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)).

EXAMPLE

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

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;

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

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Last modified December 19 02:36 EST 2018. Contains 318245 sequences. (Running on oeis4.)