A194624 Decimal expansion of the smaller solution to x^x = 3/4.

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

Offset

0,2

Comments

Since (1/e)^(1/e) < 3/4 < 1, the equation x^x = 3/4 has two solutions x = a and x = b with 0 < a < 1/e < b < 1. Both solutions are transcendental (see Proposition 2.2 in Sondow-Marques 2010).

Links

Table of n, a(n) for n=0..99.

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

Index entries for transcendental numbers

Example

0.15351678966395294715006833297846322771126948548996962031798542833437261364190...

Mathematica

x = x /. FindRoot[x^x == 3/4, {x, 0.1}, WorkingPrecision -> 120]; RealDigits[x, 10, 100] // First

Crossrefs

Cf. A030798 (x^x = 2), A072364 ((1/e)^(1/e)), A194625 (larger solution to x^x = 3/4).

Sequence in context: A071050 A271780 A176036 * A239805 A333236 A270915

Adjacent sequences: A194621 A194622 A194623 * A194625 A194626 A194627

Keyword

nonn,cons

Author

Jonathan Sondow, Sep 02 2011

Status

approved