Comment by pierrebai

Interestingly, there is a common pattern for fixed-points and cycles for different number lengths: numbers made of 4, 5 and 9. For example

length 3: fixed-point 495 length 5: 2-cycle containing 59994 length 6: fixed-point 59994

Similarly for digits 6, 1, 4, 7:

length 4: 6174 length 5: 4-cycle containing 61974 length 6: fixed-point 631764