A058891 a(n) = 2^(2^(n-1) - 1).

1, 2, 8, 128, 32768, 2147483648, 9223372036854775808, 170141183460469231731687303715884105728, 57896044618658097711785492504343953926634992332820282019728792003956564819968

Offset: 1

Author

N. J. A. Sloane, Jan 08 2001

Keywords

• nonn
• easy

Comments

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]

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

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

For n > 2, a(n) ends with 8. - Bernard Schott, Oct 20 2021

a(n) is the number of labeled hypergraphs on n - 1 vertices. - Lorenzo Sauras Altuzarra, Apr 01 2023

Examples

			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
		

References

• F. Harary, Graph Theory, Page 209, Problem 16.11.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..12
• Wikipedia, Hypergraph.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

a(n) = { 2^(2^(n-1)-1) } \\ Harry J. Smith, Jun 23 2009

Python

def A058891(n): return 1<<(1<Chai Wah Wu, Dec 12 2022

Formula

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

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

a(1) = 1, a(n+1) = 2^n*a(1)*a(2)*...*a(n). - Benoit Cloitre, Sep 13 2003

a(n) = (-1/2)*((1 + sqrt(-3))^(2^n) + (1 - sqrt(-3))^(2^n)). - Artur Jasinski, Oct 11 2008

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

a(n) = A077585(n-1) + 1. - Maurizio De Leo, Feb 25 2015

a(n) = 2^A000225(n-1). - Michel Marcus, Aug 19 2020

Sum_{n>=0} 1/a(n) = A076214. - Amiram Eldar, Oct 27 2020