

A058891


a(n) = 2^(2^(n1)  1).


75



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

1,2


COMMENTS

For n > 1, a(n) is the least solution > 1 to rad(x)^(n1) = 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^(n1) players, taking account of all matches in the tournament.  Martin Griffiths, Mar 26 2009
Also, a(n+1) = 2^(2^n1) 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


REFERENCES

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


LINKS

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


FORMULA

a(n) = A053287(A000079(n1)).
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(n1)^2 is an example with a(1) = 1 and k = 2 of a(n) = k*a(n1)^2; general explicit formula: a(n) = ((a(1)*k)^(2^(n1)))/k.  Andreas Pfaffel (andreas.pfaffel(AT)gmx.at), Apr 27 2010
a(n) = A077585(n1) + 1.  Maurizio De Leo, Feb 25 2015
a(n) = 2^A000225(n1).  Michel Marcus, Aug 19 2020
Sum_{n>=0} 1/a(n) = A076214.  Amiram Eldar, Oct 27 2020


MAPLE

a[1]:=1: for n from 2 to 20 do a[n]:=2*a[n1]^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 *)


PROG

(PARI) { t=1; for (n = 1, 12, write("b058891.txt", n, " ", 2^(t1)); t*=2; ) } \\ Harry J. Smith, Jun 23 2009


CROSSREFS

Cf. A000079, A000225, A053287, A076214, A077585, A348332.
Sequence in context: A307124 A111179 A178173 * A274171 A184945 A058343
Adjacent sequences: A058888 A058889 A058890 * A058892 A058893 A058894


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Jan 08 2001


STATUS

approved



