Sum divides product

[N. J. A. Sloane]

The issue of the Amer. Math. Monthly that arrived
today , vol 108, no. 7, Aug 2001, has the solution
to a problem which studies the sequence f(n). (On pages 668-669, Problem 10745)

f(n) is the number of solutions (r,s,t) in positive integers to the equation
 
           rst = n(r+s+t)

Could someone work out the first few terms and see if it is in the database?

Actually there are two versions, depending on whether order matters.

For n=1 we want to solve rst = r+s+t and there is
only one obvious solution, (1,2,3), or 6 solutions if order matters,
namely 123, 132, ..., 321

NJAS