OFFSET
1,2
COMMENTS
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.
LINKS
G. Marsaglia, A. Zaman and J. Marsaglia (1989), Numerical Solution of Some Classical Differential-Difference Equations, Mathematics of Computation, 53 (187), 191-201.
Jeremy Tan, Python program
J. van de Lune and E. Wattel (1969), On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory, Mathematics of Computation, 23 (106), 417-421.
Eric Weisstein's World of Mathematics, Dickman Function
FORMULA
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).
EXAMPLE
The asymptotic density of fifth-root-smooth numbers is F(1/5) = 0.000354724700... = 1/2819.08758..., so a(5) = 2819.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeremy Tan, Aug 11 2019
STATUS
approved