On the numbers divisible by all the Integers not exceeding their $r^{th}$ roots. Consider the set of all numbers which are divisible by all natural numbers not exceeding their square root, and denote this set by $S_2=\\{1,2,3,4,6,8,12,24\\}$ (Here the subscript indicates that we're taking the 2nd root of the numbers). Thus $|S_2|=8$. Similarly, the set of all numbers which are divisible by all natural numbers not exceeding the cube root is $S_3 = \\{1,2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,26,30,36,42,48,54,60,72,84,96,108,120,‌​180,240,300,420\\}$, with $|S_3|=32$. Now define $S_r$ similarly as the set of all positive numbers divisible by all the naturals not exceeding their $r^{th}$ roots. Then I have the folowing questions: **Q-1** What is the general formula for finding $|S_r|$ (ie. Cardinality of $S_r$)? **Q-2** Is there an expression for the greatest element of $S_r$? Asymptotics will also be encouraged.

I don't have asymptotics, but I coded this quick program


a(n)=my(s,old,cur,mult=1,k);while(mult<=2*(cur=(k+++1)^n-1)+9,mult=lcm(mult,k);s+=cur\mult-old\mult;old=cur);s


in PARI/GP and it finds $|S_1|=2, |S_2|=8, |S_3|=32, |S_4|=149, \ldots, |S_{1000}|\approx1.8077655\cdot10^{2571}$

which suggests that $|S_n|$ grows at roughly a factorial rate.