A194623 -id:A194623 - cp's OEIS frontend

A194622 Decimal expansion of x with 0 < x < y and x^y = y^x = 17.

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

Offset: 1

Jonathan Sondow, Aug 30 2011

Given z > 0, there exist positive real numbers x < y with x^y = y^x = z, if and only if z > e^e. In that case, (x,y) = ((1 + 1/t)^t,(1 + 1/t)^(t+1)) for some t > 0. For example, t = 1 gives 2^4 = 4^2 = 16 > e^e. When x^y = y^x = 17, at least one of x and y is transcendental. See Sondow and Marques 2010, pp. 155-157.

			x=1.7838142517704619219012767113132837917073658346795118208782477687564285462224371...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, 37 (2010), 151-164.

Cf. A073226 (e^e), A194556 ((9/4)^(27/8) = (27/8)^(9/4)), A194557 (sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3)), A194623 (y with 0 < x < y and x^y = y^x = 17).

x[t_] := (1 + 1/t)^t; y[t_] := (1 + 1/t)^(t + 1); t = t/. FindRoot[ x[t]^y[t] == 17, {t, 1}, WorkingPrecision -> 120]; RealDigits[ x[t], 10, 100] // First

A329458 Decimal expansion of x such that x^x * log(x) - x^x + 1 = 0, x > 1.

Original entry on oeis.org

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

Offset: 1

Rick Novile, Nov 16 2019

Equivalent to the coordinates of the self-intersection point of the graph y^x - x^y = y - x, where x, y > 1.

			x = 2.41173993056055928114518919802446413261177356034046370153515467138607...

Cf. A073226 (e^e), A194622, A194623, A194556, A194557.

FindRoot[x^x Log[x] - x^x + 1 == 0, {x, 2.40737, 2.41474}, WorkingPrecision -> 1000]

solve(x=2, 3, x^x * log(x) - x^x + 1) \\ Michel Marcus, Nov 16 2019

solve(x=2, 3, x - exp(1-1/x^x)) \\ Michel Marcus, Jul 14 2020

x^x * log(x) - x^x + 1 = 0; x != 1.
y^x - x^y = y - x; y = x; x != 1.
x = exp(1-x^(-x)); x > 1. - Rick Novile, Jul 14 2020

cp's OEIS Frontend

A194622 Decimal expansion of x with 0 < x < y and x^y = y^x = 17.

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