Puzzle 1080 sigma3(x) = y

Fred Schneider sent the following nice puzzle

Let's define sigma3(x) as the the sum of the cubes of the divisors of x.

1, 7401260364550416 ...

The nth term in the sequence (y = a(n)) is the minimal y such that sigma3(x) = y

has n solutions. (I have checked through 200M and have not found a 3rd term.)

The solution for a(1) is just 1 itself (because 1^3 = 1) .

The solution set for a(2) is:

194315 = 5 * 11 * 3533
184926 = 2 * 3 * 7^2 * 17 * 37

Here divisors, means all the divisors including 1 and itself.

For instance,

The divisors of 184926 are: => [1,2,3,6,7,14,17,21,34,37,42,49,51,74,98,102,111,119,147,222,238,259,294,357,518,629,714,777,
833,1258,1554,1666,1813,1887,2499,3626,3774,4403,4998,5439,8806,10878,13209,26418,30821,
61642,92463,184926]

The divisors of 194315 are: => [1,5,11,55,3533,17665,38863,194315]

And 7401260364550416=1^3+2^3+...+184926^3 = 1^3+5^3+...+194315^3

Q. Can you find more terms, the third, the fourth,...?