A185261 -id:A185261 - cp's OEIS frontend

A185262 Decimal expansion of 2^(1/phi).

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 25 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A185261

RealDigits[N[2^(1/GoldenRatio),200]][[1]]

2^((sqrt(5)-1)/2) \\ Charles R Greathouse IV, Oct 01 2012

A335020 Decimal expansion of (1/phi)^(1/phi), where phi is the golden ratio (1 + sqrt(5))/2 (A001622).

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, May 19 2020

The real function f(x) = (1/phi)^(1/phi) x^phi satisfies the differential equation f'(x) = f^(-1)(x): the derivative of f equals the compositional inverse of f.

			0.7427429446246816413695660476057885141497552527...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dr Peyam, Derivative equals inverse, youtube video (2018)

Cf. A001622, A094214, A185261.

g:= (phi-> (1/phi)^(1/phi))((1+sqrt(5))/2):
evalf(g, 140);

RealDigits[(1/GoldenRatio)^(1/GoldenRatio), 10, 100][[1]] (* Amiram Eldar, May 21 2020 *)

my(x=(sqrt(5)-1)/2); x^x \\ Michel Marcus, May 21 2020

Equals (phi-1)^(phi-1), with phi = (1 + sqrt(5))/2.

Equals A094214^A094214.

A348359 Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Timothy L. Tiffin, Oct 14 2021

The x-th root of x equals the phi-th root of phi: x^(1/x) = phi^(1/phi) = A185261 = 1.3463608200348694434247534661858... .

Not surprisingly, x appears to be irrational. If x is also algebraic, then x^phi would be transcendental by the Gelfond-Schneider theorem.

			6.055722091024741002126639117583149731683828...
x^phi = phi^x  = 18.431940924839652158136364051482054378959672... .

Wikipedia, Gelfond-Schneider theorem

Cf. A001622 (phi), A094214 (1/phi), A185261 (phi^(1/phi)), A073226 (e^e, see first comment).

RealDigits[x/.FindRoot[x^GoldenRatio==GoldenRatio^x,{x,6},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Dec 09 2024 *)

Prior Mathematica program replaced by Harvey P. Dale, Dec 09 2024

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A185262 Decimal expansion of 2^(1/phi).

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A335020 Decimal expansion of (1/phi)^(1/phi), where phi is the golden ratio (1 + sqrt(5))/2 (A001622).

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A348359 Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.

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