cp's OEIS Frontend

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

%I A309638 #13 Feb 16 2025 08:33:55
%S A309638 1,3,21,204,2819,50891,1143423,30939931,984011503,36098843631,
%T A309638 1504934136432,70436763188525,3664092112471681,210056231435360023,
%U A309638 13175390260774094846,898537704166507324228,66265550246147429710863,5259409287834480235626661,447341910388133084658686126,40620967386538406952534036284
%N A309638 Nearest integer to 1/F(1/x), where F(x) is the Dickman function.
%C A309638 The asymptotic density of the n-th-root-smooth numbers is approximately 1/a(n).
%C A309638 Van de Lune and Wattel show a(n) >= A001147(n) for n >= 1.
%H A309638 G. Marsaglia, A. Zaman and J. Marsaglia (1989), <a href="https://doi.org/10.1090/S0025-5718-1989-0969490-3">Numerical Solution of Some Classical Differential-Difference Equations</a>, Mathematics of Computation, 53 (187), 191-201.
%H A309638 Jeremy Tan, <a href="https://gitlab.com/parclytaxel/Dounreay/blob/fec7af8499c5a3cf4ba3912789a4cb0d482fa644/dickman/dickman.py">Python program</a>
%H A309638 J. van de Lune and E. Wattel (1969), <a href="https://doi.org/10.1090/S0025-5718-1969-0247789-3">On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory</a>, Mathematics of Computation, 23 (106), 417-421.
%H A309638 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DickmanFunction.html">Dickman Function</a>
%F A309638 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).
%F A309638 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).
%e A309638 The asymptotic density of fifth-root-smooth numbers is F(1/5) = 0.000354724700... = 1/2819.08758..., so a(5) = 2819.
%Y A309638 F(1/2) = A244009; F(1/3) = A175475; F(1/4) = A245238.
%K A309638 nonn
%O A309638 1,2
%A A309638 _Jeremy Tan_, Aug 11 2019