

A121296


Descending dungeons: like A121295 but read subscripts from top downwards.


12



10, 11, 13, 16, 20, 28, 45, 73, 133, 348, 4943, 22779, 537226, 11662285, 46524257772, 1092759075796059, 159271598072111595659, 3317896028408943302861454961, 594387514787460257685718548861374076357, 91930654519343922607883279072515432244874866615525276
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OFFSET

10,1


COMMENTS

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)).


EXAMPLE

a(13) = ((13_12)_11)_10 = (15_11)_10 = 16_10 = 16.


MAPLE

asubb := proc(a, b) local t1; t1:=convert(a, base, 10); add(t1[j]*b^(j1), 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 n10 do t1:=asubb(t1, ni); od: s2:=[op(s2), t1]; od;


CROSSREFS

Cf. A121263, A121265, A121295.
Sequence in context: A290745 A121263 A121295 * A121265 A045986 A216836
Adjacent sequences: A121293 A121294 A121295 * A121297 A121298 A121299


KEYWORD

nonn,base


AUTHOR

David Applegate and N. J. A. Sloane, Aug 25 2006


STATUS

approved



