sum divides product

[N. J. A. Sloane]

Thanks to Jud McCranie, Vladeta Jovovic and David Wilson,
who contributed to finishing up this problem
-at least as far as the database is concerned.
(The rate of growth is still an open problem, it seems
- see the Monthly reference)

The main sequence is this one, a new seq.:

%I A063520
%S A063520 1,3,6,5,8,8,8,14,13,9,14,17,8,18,23,18,14,17,13,33,23,10,19,36,15,22,
%T A063520 32,22,19,26,17,39,24,18,50
%N A063520 Number of solutions (r,s,t), r>=s>=t>0, to the equation rst = n(r+s+t).
%C A063520 Number of solutions (r,s) in positive integers to the equation rs = n(r+s) is tau(n^2), cf. A048691. Number of solutions (r,s), r>=s>0, to the equation rs = n(r+s) is (tau(n^2)+1)/2, cf. A018892.
%D A063520 M. J. Pelling, "The Sum Divides the Product", Problem 10745, Amer. Math. Monthly, vol. 106 (1999), p. 587; vol 108, (no. 7, Aug. 2001), pp. 668-669. [Gives upper bound]
%e A063520 There are 8 such solutions to rst = 5(r+s+t): (5, 4, 3), (7, 5, 2), (10, 4, 2), (11, 10, 1), (15, 8, 1), (20, 7, 1), (25, 3, 2), (35, 6, 1).
%Y A063520 Cf. A018892, A004194.
%K A063520 more,nonn,new
%O A063520 1,2
%A A063520 Vladeta Jovovic (vladeta at Eunet.yu), Aug 01 2001

and dww supplied more terms and 3 lines of the second
version:

(quote)
> 
> 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?

A063520 extends to

1 3 6 5 8 8 8 14 13 9 14 17 8 18 23 18 14 17 13 33 23 10 19 36 15 22 32
22 19 26 17 39 24 18 50 45 8 22 39 38 22 27 13 50 45 16 27 52 24 39 38
27 20 50 45 72 24 12 31 58 15 28 69 45 49 39 12 52 40 33 33 66 12 33 64

> 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

If order doesn't matter, we get

6 15 28 30 48 45 45 78 75 54 84 94 48 105 132 105 84 99 78 189 138 60
111 210 90 132 184 129 114 153 102 228 141 105 294 267 48 132 234 228
132 159 78 300 270 96 159 301 144 231 228 162 120 297 270 429 144 72

(end of quote)

NJAS