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User: Flavio Niccolò Baglioni

Flavio Niccolò Baglioni has authored 2 sequences.

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

A356805 Decimal expansion of the unique positive real root of the equation x^x^(x - 1) = x + 1.

Original entry on oeis.org

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

Offset: 1

Marco Ripà and Flavio Niccolò Baglioni, Aug 28 2022

This constant arises from a well-known linear approximation for real height of the tetration x^^x (for x belonging to (1, 2)), where x^^x indicates the tetration of the real base x having the same height (see Links - Wikipedia).

A valuable method to extend tetration to real numbers, and solving equations as the above, has been introduced in 2006 by Hooshmand in his paper "Ultra power and ultra exponential functions" (see Links - Hooshmand).

			1.85566023196173...

Mohammad Hadi Hooshmand, Ultra power and ultra exponential functions, Integral Transforms and Special Functions, Volume 17(8), 2006, pp. 549-558.
Wikipedia, Tetration (see in particular "Linear approximation for real heights").

Cf. A124930, A356562.

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

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

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User: Flavio Niccolò Baglioni

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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