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

1,2

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

Table of n, a(n) for n=1..20.

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

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

Sequence in context: A199682 A348912 A379186 * A167872 A380407 A192314

Adjacent sequences: A309635 A309636 A309637 * A309639 A309640 A309641

Keyword

nonn

Author

Jeremy Tan, Aug 11 2019

Status

approved