A309638 Nearest integer to 1/F(1/x), where F(x) is the Dickman function.

1, 3, 21, 204, 2819, 50891, 1143423, 30939931, 984011503, 36098843631, 1504934136432, 70436763188525, 3664092112471681, 210056231435360023, 13175390260774094846, 898537704166507324228, 66265550246147429710863, 5259409287834480235626661, 447341910388133084658686126, 40620967386538406952534036284

Offset: 1

Author

Jeremy Tan, Aug 11 2019

Keywords

• nonn

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.

Examples

			The asymptotic density of fifth-root-smooth numbers is F(1/5) = 0.000354724700... = 1/2819.08758..., so a(5) = 2819.
		

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

Crossrefs

F(1/2) = A244009; F(1/3) = A175475; F(1/4) = A245238.

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).