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I've found a number which is satisfies conditions similar to those for 381654729 but with a more orderly progression of digits:
111111222222333333444444555555666666777777888888999999
111111 is divisible by 1, and 111111222222 by 2, 111111222222333333 by 3, 111111222222333333444444 by 4 all the way up to that number above, which is divisible by 9.
Moreover the divisors can also consist of six digits: 111111222222 is divisible by 222222; 111111222222333333 by 333333 and so on.
So far I've only got this to work with a dividend consisting of each digit repeated six times, and divisors consisting of either one or six repeated digits.
(PS I sent this direct to James Grime as well, who said it was nice. The highest compliment)
Chris G