A050923 a(n) = 2^(n!).

2, 2, 4, 64, 16777216, 1329227995784915872903807060280344576

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999

Keywords

• easy
• nonn

Comments

For n > 0, every n-fold repetition of a(n) is a "powerful" arithmetic progression with difference 0; e.g., for n = 4 we get a(4) = 16777216 and in the generated repeating sequence of length 4 the k-th term is a k-th power (1 <= k <= n): 16777216 = 16777216^1, 16777216 = 4096^2, 16777216 = 256^3, 16777216 = 64^4. - Martin Renner, Aug 16 2017

From Jianing Song, Jul 20 2021: (Start)

Let F_q be the finite field with q elements, then in F_a(n), 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^lcm(1,2,...,n)) = Union_{n>=0} F_A178981(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^(n!). (End)

For n >= 1, the number of digits of a(n) is A317873(n). - Martin Renner, Mar 24 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..6
• Index to divisibility sequences

Crossrefs

Cf. A000079, A000142, A092874, A100731, A178981, A317873.

Programs

Magma

[2^Factorial(n): n in [0..8]]; // Vincenzo Librandi, Dec 16 2012

Mathematica

a=2;lst={};Do[a=a^n;AppendTo[lst,a],{n,1,7}];lst (* Vladimir Joseph Stephan Orlovsky, May 26 2009 *)
Table[2^n!,{n,0,9}] (* Vincenzo Librandi, Dec 16 2012 *)

Maxima

makelist(2^(n!),n,0,5); /* Martin Ettl, Dec 27 2012 */

PARI

a(n)=2^n! \\ Charles R Greathouse IV, Aug 16 2017

Formula

a(n) = a(n-1)^n, a(0)=2.

a(n) = A000079(A000142(n)).

Sum_{n>=1} 1/a(n) = A092874. - Amiram Eldar, Oct 27 2020