Generate all 4-perfect numbers Your program or function should output all the 36 4-perfect numbers in increasing order  separated by newlines.
An n positive integer is a k-perfect number (multiply perfect number) if the sum of its divisors including itself (i.e. the sigma function of n) equals k*n.
For example 120 is a 3-perfect number because 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360 = 3*120.
Details

Your entry should actually produce the output and terminate on your machine before you submit your answer. (i.e. the program should terminate in a reasonable time)
Hardcoding is allowed.
Built-in functions or data closely related to the problem (e.g. sigma function) are disallowed.
This is code-golf but as the task is non-trivial without major
hardcoding non-golfed entries are welcomed too.

(This is a great detailed page about the history of all multiply perfect numbers.)
Prime factorization (Ruby ,570 542 bytes)

Rev 0

Below is a table of the prime factorizations of the 36 required numbers (copied from Wolfram, hopefully with no errors.) Only 31 prime factors are involved if I have counted correctly, and some of these always occur together, for example 137*547*1093 and 683*2731*8191.

It should be possible to condense the information in the table, or perhaps even employ a search strategy. This should beat simply compressing the numbers.

Unfortunately the numbers are too big even for a 128-bit integer, which means I can't use C. Ruby and Python do support arbitrary precision integers. I have put the table in valid Ruby syntax, which makes it my first ever Ruby program! 

puts 2**5  *3**3  *5    *7
puts 2**3  *3**2  *5    *7           *13
puts 2**2  *3**2  *5    *7**2        *13      *19
puts 2**9  *3**3  *5          *11                         *31   
puts 2**7  *3**3  *5**2                  *17              *31
puts 2**9  *3**2        *7    *11    *13                  *31
puts 2**8  *3     *5    *7                    *19             *37       *73
puts 2**10 *3**3  *5**2                               *23 *31                 *89
puts 2**13 *3**3  *5          *11                                 *43                 *127
puts 2**14 *3     *5    *7                    *19         *31                                   *151
puts 2**13 *3**2        *7    *11    *13                          *43                 *127
puts 2**5  *3**4        *7**2 *11**2          *19**2                                  *127
puts 2**8  *3**2        *7**2        *13      *19**2          *37       *73           *127
puts 2**7  *3**10 *5                     *17          *23                        *107                                               *3851   
puts 2**7  *3**6  *5                     *17          *23                                  *137           *547           *1093
puts 2**14 *3**2        *7**2        *13      *19**2      *31                         *127      *151
puts 2**5  *3**4        *7**2 *11**2          *19**4                                            *151                *911
puts 2**9  *3**4        *7    *11**3                      *31**2     *61    *83                      *331
puts 2**8  *3**2        *7**2        *13      *19**4            *37     *73                     *151                *911
puts 2**25 *3**3  *5**2                       *19         *31                                                  *683            *2731       *8191
puts 2**17 *3**10       *7                    *19**2  *23       *37     *73      *107 *127                                           *3851
puts 2**17 *3**6        *7                    *19**2  *23       *37     *73           *127 *137           *547           *1093
puts 2**25 *3**4        *7    *11**2          *19**2                                  *127                     *683            *2731       *8191
puts 2**25 *3**5        *7**2        *13      *19**2                                  *127                     *683            *2731       *8191
puts 2**17 *3**10       *7                    *19**4  *23       *37     *73      *107           *151                *911             *3851
puts 2**25 *3**10 *5                          *19     *23                        *107                          *683            *2731 *3851 *8191
puts 2**17 *3**6        *7                    *19**4  *23       *37     *73                *137 *151      *547      *911 *1093
puts 2**25 *3**6  *5                          *19     *23                                  *137           *547 *683      *1093 *2731       *8191
puts 2**25 *3**4        *7    *11**2          *19**4                                            *151           *683 *911       *2731       *8191
puts 2**25 *3**5        *7**2        *13      *19**4                                            *151           *683 *911       *2731       *8191
puts 2**33 *3**4        *7    *11**3                      *31        *61    *83                      *331                                        *43691 *131071
puts 2**33 *3**10       *7    *11                     *23                   *83  *107                *331                      *3851             *43691 *131071
puts 2**33 *3**6        *7    *11                     *23                   *83            *137      *331 *547           *1093                   *43691 *131071
puts 2**37 *3**4        *7    *11**3                      *31        *61    *83                      *331                                        *43691        *174763*524287
puts 2**37 *3**10       *7    *11                     *23                   *83  *107                *331                      *3851             *43691        *174763*524287
puts 2**37 *3**6        *7    *11                     *23                   *83            *137      *331 *547           *1093                   *43691        *174763*524287


Rev 1
Just starting to gather some terms.

#683*2731*8191=15278451143, 137*547*1093=81908327, 107*3851=412057, 174763*524287=91625968981, 19**2*151*911=49659521, 83*331=27473, 11**3*31*61=2516921,37*73=2701,19*19=361

puts 2**5  *3**3  *5       *7
puts 2**3  *3**2  *5       *7           *13
puts 2**2  *3**2  *5       *7**2        *13      *19
puts 2**9  *3**3  *5             *11                         *31    
puts 2**7  *3**3  *5**2                     *17              *31
puts 2**9  *3**2           *7    *11    *13                  *31
puts 2**8  *3     *5       *7                    *19                *2701
puts 2**10 *3**3  *5**2                                  *23 *31                 *89
puts 2**13 *3**3  *5             *11                                 *43                 *127
puts 2**14 *3     *5       *7                    *19         *31                               *151
puts 2**13 *3**2           *7    *11    *13                          *43                 *127
puts 2**5  *3**4           *7**2 *11**2          *19**2                                  *127
puts 2**8  *3**2           *7**2        *13      *19**2            *2701                 *127
puts 2**7  *3**10 *5                        *17          *23                                                                           *412057  
puts 2**7  *3**6  *5                        *17          *23                                                                *81908327
puts 2**14 *3**2           *7**2        *13      *19**2      *31                         *127  *151
puts 2**5  *3**4           *7**2 *11**2          *19**2                                            *49659521
puts 2**9  *3**4           *7                                *31    *2516921  *27473
puts 2**8  *3**2           *7**2        *13      *19**2            *2701                           *49659521
puts 2**25 *3**3  *5**2                          *19         *31                                                            *15278451143
puts 2**17 *3**10          *7                    *19**2  *23       *2701                 *127                                           *412057
puts 2**17 *3**6           *7                    *19**2  *23       *2701                 *127                               *81908327
puts 2**25 *3**4           *7    *11**2          *19**2                                  *127                                                   *15278451143
puts 2**25 *3**5           *7**2        *13      *19**2                                  *127                                                   *15278451143
puts 2**17 *3**10          *7                    *19**2  *23       *2701                           *49659521                            *412057
puts 2**25 *3**10 *5                             *19     *23                                                                            *412057 *15278451143
puts 2**17 *3**6           *7                    *19**2  *23       *2701                           *49659521                *81908327
puts 2**25 *3**6  *5                             *19     *23                                                                *81908327           *15278451143
puts 2**25 *3**4           *7    *11**2          *19**2                                            *49659521                                    *15278451143
puts 2**25 *3**5           *7**2        *13      *19**2                                            *49659521                                    *15278451143
puts 2**33 *3**4           *7                                       *2516921   *27473                                                               *43691 *131071
puts 2**33 *3**10          *7    *11                     *23                   *27473                                             *412057           *43691 *131071
puts 2**33 *3**6           *7    *11                     *23                   *27473                                       *81908327               *43691 *131071
puts 2**37 *3**4           *7                                       *2516921   *27473                                                               *43691        *91625968981
puts 2**37 *3**10          *7    *11                     *23                   *27473                                             *412057           *43691        *91625968981
puts 2**37 *3**6           *7    *11                     *23                   *27473                                       *81908327               *43691        *91625968981


Rev 2, 570  542 bytes (thanks to Martin for the improvement)

As this is my first Ruby program, I just stuck with the simple approach of continuing to gather terms. Compression of the numbers would be great if anyone knows how to do that. 

I could save a few more bytes by evaluating some of the literal-only expressions in the variable assignment part, but I decided to leave them for clarity.

e=361
f=e*e*137561
g=27473
h=2516921
j=2701
k=81908327*729
m=412057*59049
n=15278451143*2**25
p=g*43691*2**33*7
q=131071
r=91625968981*16
s=127
t=1024
w=250240
x=1467648
y=7439488
puts [30240,32760,2178540,23569920,45532800,142990848,510720*j,t*42833475,t*510840*s,t*149417520,t*3099096*s,15367968*e*s,x*e*j*s,w*m,w*k,t*429378768*e*s,15367968*f,8999424*g*h,x*j*f,n*397575,t*m*y*j*s,t*k*y*j*s,n*68607*e*s,n*154791*e*s,t*20608*j*f*m,2185*n*m,t*20608*j*f*k,2185*k*n,68607*f*n,154791*f*n,81*h*p*q,253*p*m*q,253*p*k*q,81*h*p*r,253*p*m*r,253*p*k*r]