%I #75 Apr 09 2023 08:07:44

%S 1,2,8,128,32768,2147483648,9223372036854775808,

%T 170141183460469231731687303715884105728,

%U 57896044618658097711785492504343953926634992332820282019728792003956564819968

%N a(n) = 2^(2^(n-1) - 1).

%C For n > 1, a(n) is the least solution > 1 to rad(x)^(n-1) = tau(x) where rad(x) = A007947(x) is the squarefree kernel of x and tau(x) = A000005(x) the number of divisors of x. - _Benoit Cloitre_, Apr 18 2002 [Corrected by _Michel Marcus_, Oct 15 2018]

%C For n > 1, a(n) is also the total number of possible outcomes of a knockout tournament starting with 2^(n-1) players, taking account of all matches in the tournament. - _Martin Griffiths_, Mar 26 2009

%C Also, a(n+1) = 2^(2^n-1) for n >= 1 are solutions x = y of the Diophantine equation x^y * y^x = (x+y)^z in positive integers; corresponding solutions z are in A348332 (see this last sequence for more informations and links). - _Bernard Schott_, Oct 13 2021

%C For n > 2, a(n) ends with 8. - _Bernard Schott_, Oct 20 2021

%C a(n) is the number of labeled hypergraphs on n - 1 vertices. - _Lorenzo Sauras Altuzarra_, Apr 01 2023

%D F. Harary, Graph Theory, Page 209, Problem 16.11.

%H Harry J. Smith, <a href="/A058891/b058891.txt">Table of n, a(n) for n = 1..12</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>.

%F a(n) = A053287(A000079(n-1)).

%F a(1) = 1, a(n+1) = 2*a(n)^2.

%F a(1) = 1, a(n+1) = 2^n*a(1)*a(2)*...*a(n). - _Benoit Cloitre_, Sep 13 2003

%F a(n) = (-1/2)*((1 + sqrt(-3))^(2^n) + (1 - sqrt(-3))^(2^n)). - _Artur Jasinski_, Oct 11 2008

%F a(n) = 2*a(n-1)^2 is an example with a(1) = 1 and k = 2 of a(n) = k*a(n-1)^2; general explicit formula: a(n) = ((a(1)*k)^(2^(n-1)))/k. - Andreas Pfaffel (andreas.pfaffel(AT)gmx.at), Apr 27 2010

%F a(n) = A077585(n-1) + 1. - _Maurizio De Leo_, Feb 25 2015

%F a(n) = 2^A000225(n-1). - _Michel Marcus_, Aug 19 2020

%F Sum_{n>=0} 1/a(n) = A076214. - _Amiram Eldar_, Oct 27 2020

%e The 8 possible hyperedge sets for the vertex set {1, 2} are {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {2}}, {{1}, {1, 2}}, {{2}, {1, 2}} and {{1}, {2}, {1, 2}}. - _Lorenzo Sauras Altuzarra_, Apr 01 2023

%p a[1]:=1: for n from 2 to 20 do a[n]:=2*a[n-1]^2 od: seq(a[n], n=1..9); # _Zerinvary Lajos_, Apr 16 2009

%t a = 1; b = -3; Table[Expand[(-1/2) ((a + Sqrt[b])^(2^n) + (a - Sqrt[b])^(2^n))], {n, 1, 10}] (* _Artur Jasinski_, Oct 11 2008 *)

%o (PARI) { t=1; for (n = 1, 12, write("b058891.txt", n, " ", 2^(t-1)); t*=2; ) } \\ _Harry J. Smith_, Jun 23 2009

%o (Python)

%o def A058891(n): return 1<<(1<<n-1)-1 # _Chai Wah Wu_, Dec 12 2022

%Y Cf. A000079, A000225, A053287, A076214, A077585, A348332.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Jan 08 2001