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]
Divisors[n]
closely related? IsDivisible[m,n]
closely related? IsMod[m,n]
closely related? \$\endgroup\$partial
. \$\endgroup\$1598455815964665104598224777343146075218771968
is divisible by2^37
, its largest prime factor524287
is2^19-1
, and its second-largest prime factor174763 = 4*43691 - 1
, where43691
is its third-largest prime factor. \$\endgroup\$