A356810 - OEIS

%I #31 Oct 01 2022 01:17:58

%S 1,8,4,4,1,6,2,9,7,4,9,0,1,6,0,9,2,5,8,5,2,9,3,4,7,2,0,8,8,4,8,0,6,3,

%T 2,5,5,5,8,0,4,7,6,6,4,5,6,4,4,5,0,9,0,7,1,3,9,8,0,4,3,8,3,0,2,7,5,0,

%U 8,0,2,1,1,3,9,1,5,8,0,9,5,8,3,8,4,2,1,8,9,1,8,7,8,6,0,3,1,7

%N Decimal expansion of the unique root of the equation x^(x^(((log(x))^(x-1) - 1)/(log(x) - 1))) = x+1 for x in the interval [1,2].

%C This constant arises from a different interpretation of the equation x^^x = x+1, where x^^x indicates the tetration on the base x having the same height.

%C The alternative way to define x^^x is described by Takeji Ueda in his paper on Arxiv (see link below).

%C This definition implies that if Im(x) != 0, x cannot be a solution.

%C There are no other real solutions (conjecture).

%H Takeji Ueda, <a href="https://arxiv.org/abs/2105.00247">Extension of tetration to real and complex heights</a>, arXiv:2105.00247 [math.CA], 2021.

%e 1.8441629749016...

%t RealDigits[x /. FindRoot[x^(x^(((Log[x])^(x - 1) - 1)/(Log[x] - 1))) == x + 1, {x,2}, WorkingPrecision -> 100]] [[1]]

%o (PARI) solve(x=3/2, 2, x^(x^(((log(x))^(x-1) - 1)/(log(x) - 1))) - x - 1) \\ _Michel Marcus_, Aug 29 2022

%Y Cf. A356805.

%K cons,nonn

%O 1,2

%A _Flavio Niccolò Baglioni_ and _Marco Ripà_, Aug 29 2022