A085846 Decimal expansion of root of x = (1+1/x)^x.

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

Offset: 1

Author

Eric W. Weisstein, Jul 05 2003

Keywords

• nonn
• cons

Comments

Equivalently, the root of x^(x+1) = (x+1)^x.

Also a root of 1/(x^(1/x)-1) - x = 0 and 1/(x^(1/x)-1/x-1) - x = 0, which also contains the root 5.50798565277317825758902... 1/(x^(1/x)-1) ~ Pi(x) and 1/(x^(1/x)-1/x-1) ~ Pi(x), which is a much better approximation. These roots also can be computed by the recurrences x = 1/(x^(1/x)-1) and x = 1/(x^(1/x)-1/x-1). - Cino Hilliard, Sep 13 2008

This constant is transcendental (Lord, 2002). - Amiram Eldar, Oct 29 2022

Examples

			2.2931662874118610315080282912508058643722572903271212485377103961...
		

Links

• Nicolae Anghel, Foias Numbers, An. Sţiinţ. Univ. Ovidius Constanţa. Mat. (The Journal of Ovidius University of Constanţa, 2018) 26(3), 21-28.
• Nick Lord, Two Other Transcendental Numbers Obtained by (Mis)calculating e, The Mathematical Gazette, Vol. 86, No. 505 (2002), pp. 103-105.
• Eric Weisstein's World of Mathematics, Foias Constant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A021002, A124930, A169862.

Programs

Mathematica

RealDigits[ FindRoot[x^(1/x) - (x + 1)^(1/(x + 1)) == 0, {x, 2}, WorkingPrecision -> 128][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v *)

PARI

solve(x=2,3,(1+1/x)^x-x) \\ Charles R Greathouse IV, Apr 14 2014

Formula

x satisfies x^(1/x) = (x+1)^(1/(x+1)). - Marco Matosic, Nov 25 2005