OFFSET
0,1
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
FORMULA
a(n) = a(n-1)^n, a(0)=2.
Sum_{n>=1} 1/a(n) = A092874. - Amiram Eldar, Oct 27 2020
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 *)
PROG
(Magma) [2^Factorial(n): n in [0..8]]; // 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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999
STATUS
approved