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
Keywords
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
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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)
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