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Jun 8 at 0:25 comment added Gerry Myerson If the link in my comment gives the right sequence, then the next one is $107084577600=2^63^25^2\times7\times11\times13\times17\times19\times23$
Jun 7 at 22:58 comment added fleablood Oh, I misunderstood. Then.... well I actually don't undestand the problems. .... okay. Well, then it's easier. The only things less then $2\cdot....\cdot 31$ (which is $2.6$ times as big as what we've got is to increase $2$ then we have then 2 to 31. But then we want 3 times what we had with $2^3\times 2^3 \times.... 29$.
Jun 7 at 22:02 comment added Gerry Myerson As I understand it, OP insists on exponents decreasing as primes increase, so anything starting $2^3\times3\times5^2$ won't fly.
Jun 7 at 11:48 history answered fleablood CC BY-SA 4.0