TITLE: Sum of number of divisors
QUESTION [8 upvotes]: What is the value of d(1) + d(2) + d(3) + ... + d(99)? 
Here d(x) denotes the number of positive divisors of x including 1 and x.
For example, positive divisors of 4 are 1, 2, and 4 so d(4)= 3.

REPLY [7 votes]: Hint: Each number $n$ is the divisor of $n, 2n, \ldots n\lfloor \frac{99}{n} \rfloor$, and is thus counted $\lfloor \frac{99}{n} \rfloor$ times when you sum the number of divisors. Now sum over $n=1, 2, \ldots , 99$.