A050923 -id:A050923 - cp's OEIS frontend

A100731 a(n) = 3^(n!).

3, 3, 9, 729, 282429536481, 1797010299914431210413179829509605039731475627537851106401

Offset: 0

Parthasarathy Nambi, Jan 12 2005

nonn

			If n=2, 3^(2!) = 9.
If n=3, 3^(3!) = 729.
If n=4, 3^(4!) = 282429536481.

Vincenzo Librandi, Table of n, a(n) for n = 0..6

Cf. A000142, A000244, A050923.

[3^Factorial(n): n in [0..7]]; // Vincenzo Librandi, Dec 16 2012

a=3;lst={};Do[a=a^n;AppendTo[lst,a],{n,1,7}];lst (* Vladimir Joseph Stephan Orlovsky, May 26 2009 *)
Table[3^n!, {n, 0, 9}] (* Vincenzo Librandi, Dec 16 2012 *)

makelist(3^(n!),n,0,5); /* Martin Ettl, Dec 27 2012 */

From Vincenzo Librandi, Dec 16 2012: (Start)
a(n) = a(n-1)^n, a(0)=3.
a(n) = A000244(A000142(n)). (End)

A053986 a(n) = n^(n!).

Original entry on oeis.org

0, 1, 4, 729, 281474976710656, 752316384526264005099991383822237233803945956334136013765601092018187046051025390625

Offset: 0

Henry Bottomley, Apr 03 2000

Next term has 561 digits.

			a(3) = 729 because 3^3! = 3^6 = 729.

Cf. A036740, A050923, A100084.

Table[n^n!, {n, 0, 5}] (* Alonso del Arte, Jan 03 2011 *)

makelist(n^n!,n,0,6); /* Martin Ettl, Jan 13 2013 */

Sum_{n>=1} 1/a(n) = A100084. - Amiram Eldar, Nov 11 2020

One more term from Lior Manor, Nov 27 2001

A076187 Decimal expansion of Sum_{k>=0} 1/2^(k!).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Nov 02 2002

			1.2656250596046447753906250000000000007...

Cf. A050923, A092874.

digits = 105; NSum[ 1/2^k!, {k, 0, 12} , WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013 *)
RealDigits[Total[1/2^Range[0,10]!],10,120][[1]] (* Harvey P. Dale, Oct 16 2024 *)

suminf(k=0, 1/2^(k!)) \\ Michel Marcus, Feb 19 2021

Equals Sum_{k>=0} 1/A050923(k).

Equals A092874 plus 0.5. [R. J. Mathar, Sep 08 2008]

A317873 Number of digits in 2^(n!).

Original entry on oeis.org

1, 1, 1, 2, 8, 37, 217, 1518, 12138, 109238, 1092378, 12016155, 144193850, 1874520045, 26243280622, 393649209329, 6298387349264, 107072584937472, 1927306528874488, 36618824048615255, 732376480972305082, 15379906100418406713, 338357934209204947674

Offset: 0

Wolfdieter Lang and Robert G. Wilson v, Aug 09 2018

The old definition (which did not match the data) was "Number of digits in the numerators of partial sums for Liouville's constant, read as base-2 (binary) numbers (A145572)."

Kevin Ryde, Table of n, a(n) for n = 0..300

Cf. A000142, A034887, A050923, A055642, A145572.

Digits := 900: # for n <= 300
a := n -> ceil(exp(lnGAMMA(n + 1))*log10(2)):
seq(a(n), n = 0..30);  # Peter Luschny, Apr 18 2024

Mathematica
```
Array[ Floor[#! Log10@2 + 1] &, 22]
```

a(n) = A034887(n!).

Better definition suggested by Martin Renner, Mar 24 2024

a(0)=1 prepended by Alois P. Heinz, Jul 27 2025

A220078 a(n) = 5^(n!).

Original entry on oeis.org

5, 5, 25, 15625, 59604644775390625, 752316384526264005099991383822237233803945956334136013765601092018187046051025390625

Offset: 0

Vincenzo Librandi, Dec 16 2012

nonn

The next term (a(6)) has 504 digits. - Harvey P. Dale, Apr 01 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..6

Cf. A000351, A000142, A050923, A100731.

Magma
```
[5^(Factorial(n)): n in [0..7]];
```

lst={}; Do[AppendTo[lst, 5^n!], {n, 0, 9}]; lst
5^Range[6]! (* Harvey P. Dale, Apr 01 2013 *)

makelist(5^(n!),n,0,5); /* Martin Ettl, Dec 27 2012 */

a(n) = a(n-1)^n, a(0) = 5.

a(n) = A000351(A000142(n)).

One more term (a(5)) from Harvey P. Dale, Apr 01 2013

A178981 2^A003418(n); for n >= 1, the least number > 1 that can be expressed simultaneously as a k-th power of some integer for all 1 <= k <= n.

Original entry on oeis.org

2, 2, 4, 64, 4096, 1152921504606846976, 1152921504606846976

Offset: 0

Rick L. Shepherd, Jan 02 2011

Equivalently, for n >= 1, the least number > 1 of objects that can be arranged as a k-cube (k-dimensional hypercube) for all 1 <= k <= n.
a(7) = 2^420 contains 127 decimal digits.
From Jianing Song, Jul 20 2021: (Start)
Let F_q be the finite field with q elements, then F_a(n) is the smallest extension field of F_2 such that every polynomial of degree at most n splits into linear factors.
Union_{n>=0} F_a(n) is the algebraic clousre of F_2, which is the unique algebraically closed field with characteristic 2 and transcendence degree 0 (note that an algebraically closed field is uniquely determined by its characteristic and transcendence degree). Union_{n>=0} F_(2^(n!)) = Union_{n>=0} F_A050923(n) gives the same field.
Obviously, here 2 can be replaced by any prime p provided that {a(n)} is defined as a(n) = p^A003418(n). (End)

			a(6) = 2^A003418(6) = 2^60 = 1152921504606846976 [= (2^60)^1] = (2^30)^2 = 1073741824^2 = (2^20)^3 = 1048576^3 = (2^15)^4 = 32768^4 = (2^12)^5 = 4096^5 = (2^10)^6 = 1024^6, while no smaller integer > 1 can be expressed simultaneously as a square, cube, 4th, 5th, and 6th power of integers.

Jianing Song, Table of n, a(n) for n = 0..10
Index to divisibility sequences

Cf. A003418, A050923.

a(n)=2^(lcm(vector(n, i, i))) \\ Jianing Song, Jul 20 2021, following a PARI program for A003418

A203925 3^n! - 2^n!.

Original entry on oeis.org

1, 1, 5, 665, 282412759265, 1797010299914431210411850601513820123858571820477570761825

Offset: 0

Assoul Abdelkarim, Jan 08 2012

nonn

a(n) = A100731(n) - A050923(n). - Michel Marcus, Sep 04 2013

A220079 a(n) = 7^(n!).

Original entry on oeis.org

7, 7, 49, 117649, 191581231380566414401

Offset: 0

Vincenzo Librandi, Dec 16 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..6

Cf. A000142, A000420, A050923, A100731, A220078.

Magma
```
[7^Factorial(n): n in [0..7]];
```
Mathematica
```
Table[7^(n!), {n, 0, 7}]
```

makelist(7^(n!),n,0,5); /* Martin Ettl, Dec 27 2012 */

a(n) = a(n-1)^n, a(0) = 7.

a(n) = A000420(A000142(n)).

A362766 Number of nonisomorphic sets of permutations of an n-set.

Original entry on oeis.org

2, 2, 4, 24, 711936, 11076899964874307395625695676727296

Offset: 0

Andrew Howroyd, May 03 2023

nonn

Isomorphism is up to permutation of the elements of the n-set.

a(6) has 214 decimal digits.

			The a(2)=4 sets with permutations shown in cycle notation are:
  {},
  {(1)(2)},
  {(12)},
  {(1)(2), (12)}.

Andrew Howroyd, Table of n, a(n) for n = 0..6

Row sums of A362763.

Cf. A050923.

a(n) = Sum_{k=0..n!} A362763(n,k).

A050923(n) / n! <= a(n) <= A050923(n).

A305851 a(n) = (a(n-1)+a(n-2))^(n-2) with a(1)=a(2)=1.

Original entry on oeis.org

1, 1, 2, 9, 1331, 3224179360000, 348414297354956334043085401797347258376901025074126347562215651

Offset: 1

Yigithan TAMER, Jun 11 2018

a(8) has 376 digits, a(9) has 2627 digits, a(10) has 21015 digits, see b-file and a-file. - Eric Chen, Jun 14 2018

Eric Chen, Table of n, a(n) for n = 1..9
Eric Chen, Table of n, a(n) for n = 1..10 (an a-file)

Cf. A050923.

Nest[Append[#, (#[[-1]] + #[[-2]] )^(Length@ # - 2)] &, {1, 1}, 6] (* Michael De Vlieger, Jun 11 2018 *)
RecurrenceTable[{a[n] == (a[n-1] + a[n-2])^(n-2), a[1] == 1, a[2] == 1}, a, {n, 1, 7}] (* Vaclav Kotesovec, Jul 23 2018 *)
nxt[{n_,a_,b_}]:={n+1,b,(a+b)^(n-1)}; NestList[nxt,{1,1,1},7][[All,2]] (* Harvey P. Dale, Nov 03 2022 *)

a(n)=if(n<3,1,(a(n-1)+a(n-2))^(n-2)) \\ Eric Chen, Jun 14 2018

#Generates and prints a list containing the terms, up to the term with the index of seq_limit
seq_limit=9
seq_list=[1,1]
for seq_no in range(3,seq_limit):
    seq_list.append((seq_list[-1]+seq_list[-2])**(seq_no-2))
print(seq_list)

a(n) ~ c^((n-2)!), where c = 3.3203520468282576446980958620234685911457954899308569085994... - Vaclav Kotesovec, Jul 23 2018

cp's OEIS Frontend

A100731 a(n) = 3^(n!).

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Magma

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Maxima

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A053986 a(n) = n^(n!).

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Mathematica

Maxima

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A076187 Decimal expansion of Sum_{k>=0} 1/2^(k!).

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Mathematica

PARI

Formula

A317873 Number of digits in 2^(n!).

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Maple

Mathematica

Formula

Extensions

A220078 a(n) = 5^(n!).

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Programs

Magma

Mathematica

Maxima

Formula

Extensions

A178981 2^A003418(n); for n >= 1, the least number > 1 that can be expressed simultaneously as a k-th power of some integer for all 1 <= k <= n.

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PARI

A203925 3^n! - 2^n!.

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Formula

A220079 a(n) = 7^(n!).

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