%I #16 Oct 03 2021 19:25:00

%S 1,1,1,2,4,10,27,83,281,1035,4127,17656,80598,390649,2001779,10804600,

%T 61230207,363291235,2251035412,14533496547,97575061512,679975389773,

%U 4910327064257,36688562599092,283236504667511,2256366104654141,18526697776919183,156616975726597637

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

%C 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

%e a(0) = 0^0 = 1 by convention.

%e a(1) = 1, because 1^0 = 1, but any x > 0.34632336... (A333318) would make x^(1-x) > 0.5.

%e 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.

%e 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.

%t Table[First[Round[Maximize[x^(n-x),x,Reals]]],{n,0,27}] (* _Stefano Spezia_, Aug 14 2021 *)

%o (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)),", "))};

%o a346999(27)

%Y Cf. A003320.

%K nonn

%O 0,4

%A _Hugo Pfoertner_, Aug 12 2021