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A121263
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Descending dungeons: see Comments lines for definition.
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17
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10, 11, 13, 16, 20, 25, 31, 38, 46, 55, 65, 87, 135, 239, 463, 943, 1967, 4143, 8751, 18479, 38959, 103471, 306223, 942127, 2932783, 9153583, 28562479, 89028655, 277145647, 861652015, 2675637295, 10173443119, 41132125231, 168836688943, 695134284847
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OFFSET
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10,1
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COMMENTS
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Let "N_b" denote "N read in base b" and let "N" denote "N written in base 10" (as in normal life). The sequence is given by 10, 10_11, 10_(11_12), 10_(11_(12_13)), 10_(11_(12_(13_14))), etc., or in other words
......10....10.....10.....10.......etc.
..............11.....11.....11.........
.......................12.....12.......
................................13.....
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".
If a number constructed by iterating exponentials is called a "tower", perhaps these numbers should be called "dungeons".
The sequence has steady growth until a(101), but then speeds up - see the extended table. For n <= 100, a(n) grows by less than a factor of 10 each iteration. For n >= 100, a(n)/a(99) at least squares each iteration. After a(1000) it will accelerate again and so on.
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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_{i=1..n} log(9+i)), 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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For example,
10
..11
....12
......13
........14
..........15
............16
..............17
................18
..................19
....................20
......................21
........................22
..........................23
is equal to 239.
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MAPLE
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M:=100; a:=list(10..M): a[10]:=10: lprint(10, a[10]); for n from 11 to M do b:=n; for i from n-1 by -1 to 11 do t1:=convert(i, base, 10); b:=add(t1[j]*b^(j-1), j=1..nops(t1)): od: a[n]:=b; lprint(n, a[n]); od: # N. J. A. Sloane
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 # N. J. A. Sloane
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PROG
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(Python)
def a(n):
a_of_n = [((10 + int(i))) for i in range(n)]
while len(a_of_n) != 1:
exponent = 0
a_of_n [-2] = list(str(a_of_n [-2]))
for i in range(len(a_of_n [-2])):
a_of_n [-2] [-(i+1)] = int(a_of_n [-2] [-(i+1)])
a_of_n [-2] [-(i+1)] *= ((a_of_n [-1]) ** exponent)
exponent += 1
a_of_n [-2] = sum(a_of_n [-2])
a_of_n = a_of_n [:((len(a_of_n))-1)]
return (a_of_n [0])
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CROSSREFS
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KEYWORD
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nonn,nice,base
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AUTHOR
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STATUS
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approved
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