A121296 -id:A121296 - cp's OEIS frontend

A128916 a(n) = A121296(n) - A121295(n).

0, 0, 0, 0, 0, 3, 14, 35, 87, 293, 4833, 22558, 536782, 11661394, 46524255986, 1092759075792482, 159271598072111588499, 3317896028408943302861440634, 594387514787460257685718548861374047695, 91930654519343922607883279072515432244874866615467943

Offset: 10

David Applegate and N. J. A. Sloane, Sep 21 2007

nonn

All terms are >= 0.

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.

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.

A130287 a(n) = A121296(n) - A121263(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 14, 35, 87, 293, 4878, 22692, 537091, 11662046, 46524257309, 1092759075795116, 159271598072111593692, 3317896028408943302861450818, 594387514787460257685718548861374067606, 91930654519343922607883279072515432244874866615506797

Offset: 10

David Applegate and N. J. A. Sloane, Sep 21 2007

nonn

We conjecture that all terms are >= 0.

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.

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.

A121263 Descending dungeons: see Comments lines for definition.

Original entry on oeis.org

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

Offset: 10

Marc LeBrun, Aug 23 2006

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.
This is one of a family of four related sequences: alpha: A121263 (this sequence), beta: A121265, gamma: A121295, delta: A121296. The four main difference sequences are beta - alpha: A122734, beta - gamma: A127744, delta - alpha: A130287 and delta - gamma: A128916. The other two differences are gamma - alpha: A131011 and delta - beta: A131012.

			For example,
10
..11
....12
......13
........14
..........15
............16
..............17
................18
..................19
....................20
......................21
........................22
..........................23
is equal to 239.

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.

N. J. A. Sloane, Table of n, a(n) for n = 10..109
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, and N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.
Brady Haran and N. J. A. Sloane, Dungeon Numbers, Numberphile video (2020). (extra)

Cf. A121266, A121264, A121265, A121295, A121296, A121863, A121864.
Cf. A122734, A127744, A128916, A130287.
Cf. A122618 (= n_n), A121802 (the 2-adic limit of this sequence).
Cf. A049384, A124075.

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

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])
# Noah J. Crandall, Dec 07 2020

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

A121265 Descending dungeons: a(10)=10; for n>10, a(n) = a(n-1) read as if it were written in base n.

Original entry on oeis.org

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

Offset: 10

N. J. A. Sloane, Aug 23 2006

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.

			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)

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.

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 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.
Brady Haran and N. J. A. Sloane, Dungeon Numbers, Numberphile video (2020). (extra)

Cf. A121263, A121295, A121296, A127744, A122734.

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:

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

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

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

A121295 Descending dungeons: for definition see Comments lines.

Original entry on oeis.org

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

Offset: 10

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

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.
a(10) = 10; for n > 10, a(n) = n read as if it were written in base a(n-1). - Jianing Song, May 22 2021

			a(13) = 13_(12_(11_10)) = 13_(12_11) = 13_13 = 16.
From _Jianing Song_, May 22 2021: (Start)
a(10) = 10;
a(11) = 11_10 = 11;
a(12) = 12_11 = 13;
a(13) = 13_13 = 16;
a(14) = 14_16 = 20;
a(15) = 15_20 = 25;
a(16) = 16_25 = 31;
... (End)

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.

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.
Brady Haran and N. J. A. Sloane, Dungeon Numbers, Numberphile video (2020). (extra)

Cf. A121263, A121265, A121296.

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;

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

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)).

A121295(10) = 10, A121295(n) = Sum_{i=0..m-1} A121295(n-1)^(m-1-i) * d_(m-i), for n >= 11, where n = d_m,...,d_2,d_1 is the decimal expansion of n. - Christopher Hohl, Jun 11 2019

A121863 See Comments lines for definition.

Original entry on oeis.org

16, 50, 304, 93032, 17310371214, 1498244757849709540196, 3363165974015385428987990761994364730059919325224645845292529932

Offset: 4

N. J. A. Sloane, Aug 31 2006, corrected Sep 05 2006

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 16, 32_16, 64_(32_16), 128_(64_(32_16)), etc., or in other words
......16....32.....64....128.......etc.
..............16.....32.....64.........
.......................16.....32.......
................................16.....
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".
The next term is too large to include.
A "dungeon" of numbers.

			64_(32_16) = 64_(3*16 + 2) = 64_50 = 6*50 + 4 = 304.

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.

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.

Cf. A121864, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A122030.

rebase(n,bas)={ local(resul,i) ; resul= n % 10 ; i=1 ; while(n>0, n = n \10 ; resul += (n%10)*bas^i ; i++ ; ) ; return(resul) ; } { a=16 ; print(a) ; for(n=5,12, a=2^n ; forstep(j=n,5,-1, a=rebase(2^(j-1),a) ; ) ; print1(a,",") ; ) ; } \\ R. J. Mathar, Sep 01 2006

Corrected and extended by R. J. Mathar, Sep 01 2006

A121864 See Comments lines for definition.

Original entry on oeis.org

16, 50, 406, 1258008, 25465014649108, 208080288305986199465852412572946560

Offset: 4

N. J. A. Sloane, Aug 31 2006, corrected Sep 05 2006

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 16, 32_16, (64_32)_16, ((128_64)_32)_16, etc., or in other words
......16....32.....64....128.......etc.
..............16.....32.....64.........
.......................16.....32.......
................................16.....
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".
The next term is too large to include.
A "dungeon" of numbers.

			(64_32)_16 = (6*32 + 4)_16 = 196_16 = 1*256 + 9*16 + 6 = 406.

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.

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.

Cf. A121863, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A122029, A122030.

A122030 See Comments lines for definition.

Original entry on oeis.org

16, 38, 200, 2324, 1189028, 4996371438596, 54444310605180563002616118404, 1333929961635427338189657222796065386478968915403224556066936061387348857093217691444

Offset: 4

N. J. A. Sloane, Aug 31 2006

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 16, 16_32, 16_(32_64), 16_(32_(64_128)), etc., or in other words
......16....16.....16.....16.......etc.
..............32.....32.....32.........
.......................64.....64.......
................................128....
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".
The next term is too large to include.
A "dungeon" of numbers.

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.

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.

Cf. A121863, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A121864.

A124075 a(n) = 2^(3^(4^...^n)...).

Original entry on oeis.org

2, 8, 2417851639229258349412352

Offset: 2

David Applegate and N. J. A. Sloane, Nov 08 2006

The next term is too large to include.

The next term, a(5) = 2^(3^(4^5)), has 1.124...*10^488 digits. - Amiram Eldar, Jul 13 2025

			a(4) = 2^(3^4) = 2417851639229258349412352.

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.

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.

Cf. A014221, A049384, A121263, A121265, A121295, A121296.

a[n_] := Fold[#2^#1&, n, Range[2, n-1] // Reverse];
Table[a[n], {n, 2, 4}] (* Jean-François Alcover, Oct 10 2018 *)

A122029 See Comments lines for definition.

Original entry on oeis.org

16, 38, 200, 32768, 12918916616, 1242818253229988572210659846, 1900850177472859316749829932381453683166126327573485314289555274100802310696341510

Offset: 4

N. J. A. Sloane, Aug 31 2006

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 16, 16_32, (16_32)_64, ((16_32)_64)_128, etc., or in other words
......16....16.....16.....16.......etc.
..............32.....32.....32.........
.......................64.....64.......
................................128....
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".
The next term is too large to include.
A "dungeon" of numbers.

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.

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.

Cf. A121863, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A121864.

rebase(n,bas)={ local(resul,i) ; resul= n % 10 ; i=1 ; while(n>0, n = n \10 ; resul += (n%10)*bas^i ; i++ ; ) ; return(resul) ; } { a=16 ; for(p=5,10, print(a) ; a=rebase(a,2^p) ; ) ; } \\ R. J. Mathar, Sep 01 2006

Corrected and extended by R. J. Mathar, Sep 01 2006

cp's OEIS Frontend

A128916 a(n) = A121296(n) - A121295(n).

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A130287 a(n) = A121296(n) - A121263(n).

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A121263 Descending dungeons: see Comments lines for definition.

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A121265 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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A121295 Descending dungeons: for definition see Comments lines.

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A121863 See Comments lines for definition.

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A121864 See Comments lines for definition.

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A122030 See Comments lines for definition.

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