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Is the greatest common divisor of two Highly composite numbers an highly composite number? If not, has it the same number of divisors of the first lesser highly composite number?

Here below the exceptions of the first question up to the 50th highly composite number 21621600:

GCD(840, 1260) = 420 , d(420) = d(360) = 24

GCD(1260, 1680) = 420 , d(420) = d(360) = 24

GCD(720720, 1081080) = 360360 , d(360360) = d(332640) = 192

GCD(1081080, 1441440) = 360360 , d(360360) = d(332640) = 192

GCD(1081080, 2882880) = 360360 , d(360360) = d(332640) = 192

GCD(1081080, 3603600) = 360360 , d(360360) = d(332640) = 192

GCD(1081080, 7207200) = 360360 , d(360360) = d(332640) = 192

GCD(1081080, 14414400) = 360360 , d(360360) = d(332640) = 192

So, up to the 50th highly composite number, only 8 cases over 1225 (1275 if we count the trivial GCDs of two equal numbers) fail the first question but not the second. It is like if it is not an highly composite number, it is a "potential" one.

I'm actually a programmer, not a mathematician. I just happen to like math, so this could be a trivial question.

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  • $\begingroup$ Highly composite numbers are nasty beasts. I expect your empirical rule to fail at some point, but it may be out of reach of a brute force search. $\endgroup$ – Ivan Neretin Oct 4 '17 at 15:09
  • $\begingroup$ That's my fear as well. I'm gonna improve my HCN generator or find a big list to raise the bound. Up to 50th is really low indeed. $\endgroup$ – FloydCrimson Oct 4 '17 at 15:19
  • $\begingroup$ There is a pretty huge list at oeis.org/A002182 $\endgroup$ – Ivan Neretin Oct 4 '17 at 15:22

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