As @StevenStadnicki points out, it's interesting to investigate the structure of these sets. (I started to do it by hand, and found it sort of addictive.) Here's some Haskell code to allocate the first $N$ numbers (doubtless both inefficient and unidiomatic, but it seems to work):
insert N [] = [(N, [N])]
insert N ((c,bs):bss) = if gcd c N == 1 then (N*c,N:bs):bss else (c,bs):(insert N bss)
insertTo 1 = []
insertTo N = insert N $ insertTo (N - 1)
One runs it as
map snd $ insertTo 1000
(for example), whose output starts
[[997,991,983,977,971,967,953,947,941,937,929,919,911,907,887,883,881,877,863,859,857,853,839,829,827,823,821,811,809,797,787,773,769,761,757,751,743,739,733,727,719,709,701,691,683,677,673,661,659,653,647,643,641,631,619,617,613,607,601,599,593,587,577,571,569,563,557,547,541,523,521,509,503,499,491,487,479,467,463,461,457,449,443,439,433,431,421,419,409,401,397,389,383,379,373,367,359,353,349,347,337,331,317,313,311,307,293,283,281,277,271,269,263,257,251,241,239,233,229,227,223,211,199,197,193,191,181,179,173,167,163,157,151,149,139,137,131,127,113,109,107,103,101,97,89,83,79,73,71,67,61,59,53,47,43,41,37,31,29,23,19,17,13,11,7,5,3,2],
[961,841,529,361,289,169,121,49,25,9,4],
[667,323,143,35,6],[899,437,221,77,15,8],
[713,247,187,21,10],[551,391,91,55,12],
[851,493,209,65,27,14],[299,133,85,33,16],
[377,253,119,95,18],[703,527,319,161,39,20],
[989,779,629,403,203,45,22],
[893,731,533,407,217,115,24],
[943,817,341,259,125,51,26],
[799,481,451,145,57,28],
[901,611,589,473,287,30] …