Prove that for every positive integer m there exists a positive integer k, such that 2km is a digit anagram of km.
(In reply to
seems true - here are examples, but no proof by Steven Lord)
142857= 999999/7
so the cycles for 1/7, 2/7,3/7 .....are 142857, 285714, 428571 etc
The above supports several of your numbers....