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A121265
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Descending dungeons: a(10)=10; for n>10, a(n) = a(n-1) read as if it were written in base n.
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14
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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
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10,1
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COMMENTS
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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 base-b expansion".
A "dungeon" of numbers.
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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M:=35; a:=list(10..M): a[10]:=10: lprint(10, a[10]); for n from 11 to M do t1:=convert(a[n-1], base, 10); a[n]:=add(t1[i]*n^(i-1), i=1..nops(t1)); lprint(n, a[n]); od:
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MATHEMATICA
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nxt[{n_, a_}]:={n+1, FromDigits[IntegerDigits[a], n+1]}; Transpose[ NestList[ nxt, {10, 10}, 20]][[2]] (* Harvey P. Dale, Jul 13 2014 *)
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PROG
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(PARI) a(n) = {my(x=10); for (b=11, n, x = fromdigits(digits(x, 10), b); ); x; } \\ Michel Marcus, May 26 2019
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CROSSREFS
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KEYWORD
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nonn,base,nice,changed
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AUTHOR
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STATUS
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approved
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