A226776 Decimal expansion of the maximum value of f(x) = x - log(x)^log(x).

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

Offset: 1

Author

Richard R. Forberg, Jun 17 2013

Keywords

• nonn
• cons

Comments

The value of x where f(x) is maximum is 8.06157... Note that greater precision in this value is made difficult due to a broad "flat" maximum.

In the recursive formula: b(n+1) = log(b(n))^log(b(n)) + c, where c is a constant, the maximum value of c, without the recursion diverging to infinity, is maximum value of the function above (3.4172192....), with b(1) set anywhere in the range 1 < b(1) <= 8.06157.

At values of c between 3.4172192 +/- 0.0000005 demonstrable convergence or divergence of the recursion above takes tens of thousands of iterations, increasing with further closeness to 3.4172192517331..., if b(1) is within the range above.

At x = e^e = 15.1542... (see A073226), the function f(x) = 0. This is the only value for b(1) where the recursion above is stable when c = 0.

Examples

			3.4172192517331903782941430626511991141665169728869621034583784206062...
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Programs

Mathematica

digits = 92; FindMaximum[x-Log[x]^Log[x], {x, 3}, WorkingPrecision -> digits] // First // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014 *)

PARI

(x->x-log(x)^log(x))(solve(x=8,9,my(L=log(x));1-L^L*(1+log(L))/x)) \\ Charles R Greathouse IV, Jun 18 2013