cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A121866 Same as A121864 but start at 8 instead of 16.

Original entry on oeis.org

8, 14, 40, 262, 352264, 1484585652808, 86279953484628827950028778425712, 7605980144769110332172415576695326607722497051737088217125532622141954502574772559281862172625424410193559821272357826
Offset: 3

Views

Author

N. J. A. Sloane, Sep 05 2006

Keywords

Comments

The convention used here is that e.g. 9_8 = 9.
The next term is too large to include.

References

  • 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.

Links

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

Views

Author

Marc LeBrun, Aug 23 2006

Keywords

Comments

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.

Examples

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

References

  • 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.

Links

Crossrefs

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.

Programs

  • Maple
    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
  • 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])
# Noah J. Crandall, Dec 07 2020

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_{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

A121863 See Comments lines for definition.

Original entry on oeis.org

16, 50, 304, 93032, 17310371214, 1498244757849709540196, 3363165974015385428987990761994364730059919325224645845292529932
Offset: 4

Views

Author

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

Keywords

Comments

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.

Examples

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

References

  • 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.

Links

Crossrefs

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

Programs

  • PARI
    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

Extensions

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

A122030 See Comments lines for definition.

Original entry on oeis.org

16, 38, 200, 2324, 1189028, 4996371438596, 54444310605180563002616118404, 1333929961635427338189657222796065386478968915403224556066936061387348857093217691444
Offset: 4

Views

Author

N. J. A. Sloane, Aug 31 2006

Keywords

Comments

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.

References

  • 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.

Links

Crossrefs

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

A122029 See Comments lines for definition.

Original entry on oeis.org

16, 38, 200, 32768, 12918916616, 1242818253229988572210659846, 1900850177472859316749829932381453683166126327573485314289555274100802310696341510
Offset: 4

Views

Author

N. J. A. Sloane, Aug 31 2006

Keywords

Comments

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.

References

  • 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.

Links

Crossrefs

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

Programs

  • PARI
    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

Extensions

Corrected and extended by R. J. Mathar, Sep 01 2006
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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