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A346999
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a(n) is the maximum of x^(n - x), rounded to the nearest integer, for nonnegative real x.
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1
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Table[First[Round[Maximize[x^(n-x), x, Reals]]], {n, 0, 27}] (* Stefano Spezia, Aug 14 2021 *)
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PROG
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(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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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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