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

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..81.

Formula

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

Example

x = 2.41173993056055928114518919802446413261177356034046370153515467138607...

Mathematica

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

Prog

(PARI) solve(x=2, 3, x^x * log(x) - x^x + 1) \\ Michel Marcus, Nov 16 2019
(PARI) solve(x=2, 3, x - exp(1-1/x^x)) \\ Michel Marcus, Jul 14 2020

Crossrefs

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

Sequence in context: A096110 A207260 A187913 * A332010 A201287 A236367

Adjacent sequences: A329455 A329456 A329457 * A329459 A329460 A329461

Keyword

nonn,cons

Author

Rick Novile, Nov 16 2019

Status

approved