A356805 -id:A356805 - cp's OEIS frontend

A356810 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].

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, 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, 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

Offset: 1

Flavio Niccolò Baglioni and Marco Ripà, Aug 29 2022

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.
The alternative way to define x^^x is described by Takeji Ueda in his paper on Arxiv (see link below).
This definition implies that if Im(x) != 0, x cannot be a solution.
There are no other real solutions (conjecture).

			1.8441629749016...

Takeji Ueda, Extension of tetration to real and complex heights, arXiv:2105.00247 [math.CA], 2021.

Cf. A356805.

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

solve(x=3/2, 2, x^(x^(((log(x))^(x-1) - 1)/(log(x) - 1))) - x - 1) \\ Michel Marcus, Aug 29 2022

cp's OEIS Frontend

A356810 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].

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