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    $\begingroup$ I see that it is very difficult to construct at least in a multiplicative way a subset of highly abundant numbers, does one exist? $\endgroup$ Commented Mar 28 at 21:47
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    $\begingroup$ I'm not aware of the patterns, but you can search for those in the factorizations of first highly abundant numbers - OEIS provides $10^5$ of them in oeis.org/A002093/b002093.txt Also, in that range it's easy to test if a particular number is HA - if it lays strictly in between two consecutive terms, it's not HA. $\endgroup$ Commented Mar 28 at 22:38
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    $\begingroup$ Beyond that range, you can test for HAness with my SageMath code posted in mathoverflow.net/q/501164 $\endgroup$ Commented Mar 28 at 22:44
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    $\begingroup$ @JoséDamiánEspinosa see the Colossally Abundant Numbers en.wikipedia.org/wiki/Colossally_abundant_number $\endgroup$ Commented Mar 28 at 22:53
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    $\begingroup$ I wrote mathoverflow.net/questions/79927/… $\endgroup$ Commented Mar 28 at 22:57