

A121265


Descending dungeons: a(10)=10; for n>10, a(n) = a(n1) read as if it were written in base n.


14



10, 11, 13, 16, 20, 30, 48, 76, 132, 420, 1640, 11991, 249459, 14103793, 5358891675, 19563802363305, 3359230167951561129, 181335944930584275675841374, 54416647690014492928933662292768871352, 6605721238793689879501639879905020611382966457124120828, 360539645288616164606228883801608423987740093330992456820074646988075733781927268
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OFFSET

10,1


COMMENTS

Using N_b to denote "N read in base b", the sequence is given by
......10....10.....10.....10.......etc.
..............11.....11.....11.........
.......................12.....12.......
................................13.....
where the subscripts are evaluated from the top downwards.
More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a baseb expansion".
A "dungeon" of numbers.


REFERENCES

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated BaseChanging, 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. 393402.


LINKS

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 BaseChanging, arXiv:math/0611293 [math.NT], 20062007.
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466467.
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)).  David Applegate and N. J. A. Sloane, Aug 25 2006


EXAMPLE

From Jianing Song, May 22 2021: (Start)
a(10) = 10;
a(11) = 10_11 = 11;
a(12) = 11_12 = 13;
a(13) = 13_13 = 16;
a(14) = 16_14 = 20;
a(15) = 20_15 = 30;
a(16) = 30_16 = 48;
... (End)


MAPLE

M:=35; a:=list(10..M): a[10]:=10: lprint(10, a[10]); for n from 11 to M do t1:=convert(a[n1], base, 10); a[n]:=add(t1[i]*n^(i1), i=1..nops(t1)); lprint(n, a[n]); od:


MATHEMATICA

nxt[{n_, a_}]:={n+1, FromDigits[IntegerDigits[a], n+1]}; Transpose[ NestList[ nxt, {10, 10}, 20]][[2]] (* Harvey P. Dale, Jul 13 2014 *)


PROG

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


CROSSREFS

Cf. A121263, A121295, A121296, A127744, A122734.
Sequence in context: A121263 A121295 A121296 * A045986 A216836 A188165
Adjacent sequences: A121262 A121263 A121264 * A121266 A121267 A121268


KEYWORD

nonn,base,nice


AUTHOR

N. J. A. Sloane, Aug 23 2006


STATUS

approved



