A049384 -id:A049384 - cp's OEIS frontend

A080219 Decimal expansion of exponential factorial constant Sum_{n>=1} 1/A049384(n).

1, 6, 1, 1, 1, 1, 4, 9, 2, 5, 8, 0, 8, 3, 7, 6, 7, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Eric W. Weisstein, Feb 06 2003

This is a Liouville number and therefore transcendental.

			1/1 + 1/2 + 1/9 + 1/262144 + ... = 1.611114925808376736111...

Contributed by Jonathan Sondow.

J. Sondow, MathWorld: Exponential Factorial
J. Sondow, Irrationality measures, irrationality bases, and a theorem of Jarnik, arXiv:math/0406300 [math.NT], 2004; see L_4 in Example 4.
Wikipedia, Exponential factorial
Wikipedia, Liouville number
Index entries for transcendental numbers

Cf. A049384, A167155.

eFac[1] = 1; eFac[n_] := eFac[n] = n^eFac[n-1]; Clear[s]; s[m_] := s[m] = RealDigits[Sum[1/eFac[n], {n, 1, m}], 10, 100] // First; s[m = 1]; While[s[m] != s[m - 1], m++]; s[m] (* Jean-François Alcover, Feb 08 2013 *)

A138878 First differences of A049384.

Original entry on oeis.org

0, 1, 7, 262135

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 16 2008

nonn

A218583 Decimal expansion of the limit of the n-fold application of the natural logarithm to A049384 as n tends to infinity.

Original entry on oeis.org

0, 6, 1, 3, 3, 1, 2, 4, 2, 3, 0, 0, 0, 8, 3, 5, 1, 2, 3, 4, 3, 9, 8, 5, 5, 9, 9, 6, 9, 5, 0, 0, 6, 0, 4, 5, 0, 6, 1, 2, 1, 0, 2, 6, 4, 5, 2, 8, 7, 7, 9, 7, 3, 3, 6, 4, 2, 5, 7, 5, 2, 0, 6, 6, 3, 4, 7, 8, 1, 6, 6, 9, 5, 6, 2, 0, 2, 8, 9, 4, 7, 0, 1, 6, 5, 1, 8, 3, 8, 9, 2, 5, 7, 9, 7, 4, 4, 8, 2

Offset: 0

Benedikt Otten, Nov 03 2012

The value can be calculated to an accuracy of at least 4.8*10^183230 decimal digits by calculating log(...log(7^...^1)...).

			-0.0613312423000835123439855996950060450612102645287...

Proof that the series converges (StackExchange, 10/09/2012)

Cf. A049384.

p[n_] := HoldForm[n]^(p[n - 1]); p[1] := 1; rules = {Log[x_ y_] :> Log[x] + Log[y], Log[x_^k_] :> k Log[x]}; lnn[x_, n_] := Log[lnn[x, n - 1]]; lnn[x_, 0] := x; RealDigits[ReleaseHold[lnn[p[7], 7] //. rules], 10, 100, 0]

Equals -log(...log(n^(n-1)^...^1)...) (n nested log).

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

A140319 Exponential primorial: a(n)=prime(n)^a(n-1), a(0)=1, where prime(n)=A000040(n).

Original entry on oeis.org

1, 2, 9, 1953125

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 26 2008

The next term is too large to include.

			a(1) = prime(1)^a(0) = 2^1 = 2.
a(2) = 3^2 = 9.
a(3) = 5^9 = 1953125.
a(4) = 7^1953125 has 1650583 digits, starting with 12864794... and ending in ...31920807. - _M. F. Hasler_, Nov 03 2009

Jonathan Sondow, Exponential Factorial, Eric Weisstein's World of Mathematics.

Cf. A002110, A049384, A139802, A167155.

Cf. A152859 (alternate definition: start with a(0)=0), A167155.

P:=proc(n) local a,i; a:=2; print(a); for i from 2 by 1 to n do a:=ithprime(i)^a; print(a); od; end: P(5);

a[0]=1;a[n_]:=Prime[n]^a[n-1];Array[a,4,0] (* James C. McMahon, Jul 08 2025 *)

A140319(n)=if(n,prime(n)^A140319(n-1),1) \\ M. F. Hasler, Nov 03 2009

Sum_{n>=0} 1/a(n) = A167155. - Amiram Eldar, Nov 15 2020

Corrected offset/definition, and initial a(0)=1 from M. F. Hasler, Nov 03 2009

A027747 a(n) = n^n^n^..^n (6-n terms, n<=5).

Original entry on oeis.org

1, 65536, 7625597484987, 256, 5

Offset: 1

Sent in by an anonymous correspondent

The creator of the test asked me to remove this sequence. But one of the reasons for creating this database was to help people do these tests. This was stated already in the 1973 Handbook of Integer Sequences. Many sequences in the database have appeared on IQ tests.

			a(1) = 1^(1^(1^(1^1))) = 1.
a(2) = 2^(2^(2^2)) = 2^16.
a(3) = 3^(3^3) = 3^27.
a(4) = 4^4.
a(5) = 5.

P. Cooijmans, Test for Genius (Short Form) [broken link]
Paul Cooijmans, Numbers.
Paul Cooijmans, Short Test For Genius.

Cf. A004231, A049384, A051441.

b:= (n, i)-> `if`(i=0, 1, n^b(n, i-1)):
a:= n-> b(n, 6-n):
seq(a(n), n=1..5);  # Alois P. Heinz, Aug 22 2017

Examples from Martin Renner, Aug 22 2017

A067039 The tower function n^{(n-1)!}.

Original entry on oeis.org

1, 2, 9, 4096, 59604644775390625

Offset: 1

Amarnath Murthy, Dec 29 2001

nonn

a(n) = n^(n-1)^(n-2)^...^3^2^1 with all power operators nested from the left. Nesting from the right gives A049384. - Gus Wiseman, Jul 03 2019

			a(4) = 4^(3!) = 4^6 = 4096.

Cf. A000142, A007489, A049384, A052409, A052410, A140319, A294338, A316782, A325624.

makelist((n+1)^(n!),n,0,6); /* Martin Ettl, Jan 17 2013 */

A103562 a(n) = n^((n-1)^(n-2)), n>=2.

Original entry on oeis.org

2, 9, 262144, 542101086242752217003726400434970855712890625

Offset: 2

Zak Seidov, Mar 23 2005

nonn

Note that a(6) has 487 digits, cf. A049384.

The sequence starts at a(2); the continuous extension would have a(1+) = e in the limit. [Karl W. Heuer, Nov 25 2014]

Cf. A049384.

Table[n^(n-1)^(n-2),{n,2,6}] (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)

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 *)

A073581 Factorials successively exponentiated.

Original entry on oeis.org

1, 1, 2, 36, 48708493958471199415506599153950129703565945470976

Offset: 0

Jon Kongsvold (Jon.Kongsvold(AT)idi.ntnu.no), Aug 28 2002

a(5) > 10^(10^50).

			a(3) = 36 because 3!^(2!^(1!^0!)) = 36

Cf. A049384, A358972 (exponents left to right).

Mathematica
```
a[0] := 1 a[n_Integer] := n!^a[n - 1]
```

a(n) = n!^a(n - 1) starting with a(0)=0!=1.

Edited by Henry Bottomley, Jul 13 2003

cp's OEIS Frontend

A080219 Decimal expansion of exponential factorial constant Sum_{n>=1} 1/A049384(n).

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A138878 First differences of A049384.

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A218583 Decimal expansion of the limit of the n-fold application of the natural logarithm to A049384 as n tends to infinity.

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A121263 Descending dungeons: see Comments lines for definition.

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A140319 Exponential primorial: a(n)=prime(n)^a(n-1), a(0)=1, where prime(n)=A000040(n).

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A027747 a(n) = n^n^n^..^n (6-n terms, n<=5).

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Maple

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A067039 The tower function n^{(n-1)!}.

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Maxima

A103562 a(n) = n^((n-1)^(n-2)), n>=2.

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