A346999 a(n) is the maximum of x^(n - x), rounded to the nearest integer, for nonnegative real x.

1, 1, 1, 2, 4, 10, 27, 83, 281, 1035, 4127, 17656, 80598, 390649, 2001779, 10804600, 61230207, 363291235, 2251035412, 14533496547, 97575061512, 679975389773, 4910327064257, 36688562599092, 283236504667511, 2256366104654141, 18526697776919183, 156616975726597637

Offset: 0

Author

Hugo Pfoertner, Aug 12 2021

Keywords

• nonn

Comments

The maximum of f(x) = x^(n-x) occurs at x_m(n) that is the solution of x*(1+log(x)) = n. - Bernard Schott, Oct 03 2021

Examples

			a(0) = 0^0 = 1 by convention.
a(1) = 1, because 1^0 = 1, but any x > 0.34632336... (A333318) would make x^(1-x) > 0.5.
a(2) = 1 because the maximum of f(x) = x^(2-x) occurs at x_m = 1.4547332..., f(x_m) = 1.2267621..., round(f(x_m)) = 1.
a(5) = 10: maximum of f(x) = x^(5-x) occurs at x_m = 2.57141358157..., f(x_m) = 9.91146808..., round(f(x_m)) = 10.
		

Crossrefs

Cf. A003320.

Programs

Mathematica

Table[First[Round[Maximize[x^(n-x),x,Reals]]],{n,0,27}] (* Stefano Spezia, Aug 14 2021 *)

PARI

a346999(limit) = {my(d(n,y)=derivnum(x=y,x^(n-x))); print1(0^0,", "); for(n=1,limit, my(X=solve(x=1,n,d(n,x))); print1(round(X^(n-X)),", "))};
a346999(27)