These nine fixed points or cycles occur with the following frequencies (1 ⇐ n ⇐ 10e9): 33.3% : (153 → ) 29.5% : (371 → ) 17.8% : (370 → ) 5.0% : (55 → 250 → 133 → ) 4.1% : (160 → 217 → 352 → ) 3.8% : (407 → ) 3.1% : (919 → 1459 → ) 1.8% : (1 → ) 1.5% : (136 → 244 → )
No other fixed points or cycles are possible (except 0 → 0, which isn’t reachable from any nonzero input) since any number with more than four digits will have fewer digits in the sum of its cubed digits.
These nine fixed points or cycles occur with the following frequencies (1 ⇐ n ⇐ 10e9):
33.3% : (153 → )
29.5% : (371 → )
17.8% : (370 → )
5.0% : (55 → 250 → 133 → )
4.1% : (160 → 217 → 352 → )
3.8% : (407 → )
3.1% : (919 → 1459 → )
1.8% : (1 → )
1.5% : (136 → 244 → )
No other fixed points or cycles are possible (except 0 → 0, which isn’t reachable from any nonzero input) since any number with more than four digits will have fewer digits in the sum of its cubed digits.