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A309638
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Nearest integer to 1/F(1/x), where F(x) is the Dickman function.
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1
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1, 3, 21, 204, 2819, 50891, 1143423, 30939931, 984011503, 36098843631, 1504934136432, 70436763188525, 3664092112471681, 210056231435360023, 13175390260774094846, 898537704166507324228, 66265550246147429710863, 5259409287834480235626661, 447341910388133084658686126, 40620967386538406952534036284
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OFFSET
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1,2
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COMMENTS
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The asymptotic density of the n-th-root-smooth numbers is approximately 1/a(n).
Van de Lune and Wattel show a(n) >= A001147(n) for n >= 1.
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LINKS
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FORMULA
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1/F(1/x) = 1/rho(x), where rho(x) satisfies rho'(x) = -rho(x-1)/x and rho(x) = 1 for x <= 1. rho(x) may be computed to arbitrary precision by the method of Marsaglia, Zaman and Marsaglia (implemented in the Python program in Links).
a(n) ~ exp(Ei(t) - n*t) / (t * sqrt(2*Pi*n)), where Ei is the exponential integral and t is the positive root of exp(t) - n*t - 1 (van de Lune and Wattel).
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EXAMPLE
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The asymptotic density of fifth-root-smooth numbers is F(1/5) = 0.000354724700... = 1/2819.08758..., so a(5) = 2819.
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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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