If you take each of the digits of 153, cube them, and then sum those cubes, you get 153:
1 + 125 + 27 = 153.
For many naturals, if you iteratively apply this function, you’ll return to the 153 fixed point. Start with, say, 298:
8 + 729 + 512 = 1,249
1 + 8 + 64 + 729 = 802
512 + 0 + 8 = 516
125 + 1 + 216 = 342
27 + 64 + 8 = 99
729 + 729 = 1,458
1 + 64 + 125 + 512 = 702
343 + 0 + 8 = 351
27 + 125 + 1 = 153
1 + 125 + 27 = 153
1 + 125 + 27 = 153...
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.
153
If you take each of the digits of 153, cube them, and then sum those cubes, you get 153:
1 + 125 + 27 = 153.
For many naturals, if you iteratively apply this function, you’ll return to the 153 fixed point. Start with, say, 298:
8 + 729 + 512 = 1,249
1 + 8 + 64 + 729 = 802
512 + 0 + 8 = 516
125 + 1 + 216 = 342
27 + 64 + 8 = 99
729 + 729 = 1,458
1 + 64 + 125 + 512 = 702
343 + 0 + 8 = 351
27 + 125 + 1 = 153
1 + 125 + 27 = 153
1 + 125 + 27 = 153...
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.