A121263 -id:A121263 - cp's OEIS frontend

A121266 Triangle read by rows: row n (n>= 10) gives n-10 successive bases used in computing A121263(n) followed by A121263(n) itself.

10, 11, 11, 12, 13, 13, 13, 15, 16, 16, 14, 17, 19, 20, 20, 15, 19, 22, 24, 25, 25, 16, 21, 25, 28, 30, 31, 31, 17, 23, 28, 32, 35, 37, 38, 38, 18, 25, 31, 36, 40, 43, 45, 46, 46, 19, 27, 34, 40, 45, 49, 52, 54, 55, 55, 20, 29, 37, 44, 50, 55, 59, 62, 64, 65, 65

Offset: 10

N. J. A. Sloane, Aug 23 2006

Left-hand entry of row n is n, right-hand entry is A121263(n).

A "dungeon" of numbers.

			Triangle begins:
10
11 11
12 13 13
13 15 16 16
14 17 19 20 20
15 19 22 24 25 25
16 21 25 28 30 31 31
17 23 28 32 35 37 38 38
18 25 31 36 40 43 45 46 46
19 27 34 40 45 49 52 54 55 55
20 29 37 44 50 55 59 62 64 65 65

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, Rows 10 through 45 of triangle, flattened
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. A121263.

M:=45; a:=list(10..M): a[10]:=10: a[10]; for n from 11 to M do b:=n; lprint(b); 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)): lprint(b); od: a[n]:=b; lprint(a[n]); od:

A122734 a(n) = A121265(n) - A121263(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 17, 38, 86, 365, 1575, 11904, 249324, 14103554, 5358891212, 19563802362362, 3359230167951559162, 181335944930584275675837231, 54416647690014492928933662292768862601, 6605721238793689879501639879905020611382966457124102349, 360539645288616164606228883801608423987740093330992456820074646988075733781888309

Offset: 10

David Applegate and N. J. A. Sloane, Sep 24 2006

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.

A121802 The numbers A121263(n) converge 2-adically. This sequence shows their 2-adic limit.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1

Offset: 0

N. J. A. Sloane, Oct 08 2006

A121263 converges k-adically for any k which is not divisible by a prime greater than 7.

			The 2-adic expansions (that is, the binary expansions written backwards) of terms 30 through 43 of A121263 are:
30, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1]
31, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1]
32, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1]
33, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1]
34, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1]
35, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1]
36, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1]
37, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1]
38, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
39, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1]
40, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1]
41, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1]
42, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1]
43, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1]
44, [1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1]
and we can see that the initial terms are converging.

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.

A131011 a(n) = A121295(n) - A121263(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 134, 309, 652, 1323, 2634, 5193, 10184, 19911, 38854, 133040, 412527, 1241773, 3701864, 10999194, 32642353, 96825335, 287134794, 851344708, 2523819059, 10866247001, 43994094066, 175538023511, 697843906028

Offset: 10

N. J. A. Sloane, Sep 24 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.

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

A121296 Descending dungeons: like A121295 but read subscripts from top downwards.

Original entry on oeis.org

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

Offset: 10

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

A "dungeon" of numbers.

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

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, A121265, A121295.

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
s2:=[10]; for n from 11 to 35 do t1:=n; for i from 1 to n-10 do t1:=asubb(t1,n-i); od: s2:=[op(s2),t1]; od;

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)).

A121264 See Comments for definition.

Original entry on oeis.org

2, 3, 5, 26, 1370, 9840770, 851566070376026

Offset: 1

Marc LeBrun, Aug 23 2006

We use N_b to denote "N read in base b". Similar to A121263, but now we write the numerals in the stack in binary:
......10....10.....10......10........etc.
..............11.....11......11..........
.......................100.....100.......
..................................101....
The next two terms have about 144 and 450 digits respectively.
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. A121263, A121265.

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

cp's OEIS Frontend

A121266 Triangle read by rows: row n (n>= 10) gives n-10 successive bases used in computing A121263(n) followed by A121263(n) itself.

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A122734 a(n) = A121265(n) - A121263(n).

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A121802 The numbers A121263(n) converge 2-adically. This sequence shows their 2-adic limit.

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

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A131011 a(n) = A121295(n) - A121263(n).

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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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A121296 Descending dungeons: like A121295 but read subscripts from top downwards.

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