%I #52 Mar 30 2024 13:12:50
%S 2,2,4,64,16777216,1329227995784915872903807060280344576
%N a(n) = 2^(n!).
%C 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
%C From _Jianing Song_, Jul 20 2021: (Start)
%C 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.
%C 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.
%C Obviously, here 2 can be replaced by any prime p provided that {a(n)} is defined as a(n) = p^(n!). (End)
%C For n >= 1, the number of digits of a(n) is A317873(n). - _Martin Renner_, Mar 24 2024
%H Vincenzo Librandi, <a href="/A050923/b050923.txt">Table of n, a(n) for n = 0..6</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = a(n-1)^n, a(0)=2.
%F a(n) = A000079(A000142(n)).
%F Sum_{n>=1} 1/a(n) = A092874. - _Amiram Eldar_, Oct 27 2020
%t a=2;lst={};Do[a=a^n;AppendTo[lst,a],{n,1,7}];lst (* _Vladimir Joseph Stephan Orlovsky_, May 26 2009 *)
%t Table[2^n!,{n,0,9}] (* _Vincenzo Librandi_, Dec 16 2012 *)
%o (Magma) [2^Factorial(n): n in [0..8]]; // _Vincenzo Librandi_, Dec 16 2012
%o (Maxima) makelist(2^(n!),n,0,5); /* _Martin Ettl_, Dec 27 2012 */
%o (PARI) a(n)=2^n! \\ _Charles R Greathouse IV_, Aug 16 2017
%Y Cf. A000079, A000142, A092874, A100731, A178981, A317873.
%K easy,nonn
%O 0,1
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999