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There isn't so much a problem to solve here as the presentation of results.

Within the realm of multiples of three we have the following:
If you sum the cubes of the digits of such a number you get another number which is also a multiple of three.
(Mini-proof: the (1 mod 3's) cube to (1 mod 3) and the (2 mod 3's) cube to (2 mod 3) and stay in balance, since sum of digits mod 3 = 0 for multiples of 3.)

If you iterate the process of summing cube of digits, you ultimately arrive at 153, which equals the sum of its own digits. 1 + 125 + 27 = 153.
 It would be interesting to have a proof, rather than just experimental results.
For further information, see A346630 and A046197 in the Online Encyclopedia of Integer Sequences.

For numbers other than multiples of three, either a stable point is reached, or a cycle of numbers repeats.

Other fixed point numbers which are not multiples of 3 are:
370, 371, and 407 (from A046197).

0 and 1, as they often are, are special cases.

Below we display a tree of the results for numbers up to about 1600.
For example, you can find 1539 in the first column on the line marked with a #.
The cube-sum of its digits is 882, found at the head of that line.
Then, working up and to the left, you can find the sequence of cube-sums, which it generates, as shown below.

For any number you can use the auto calculator provided.
Enter the number you desire to calculate in the box and click 'calculate'. The path will be displayed.
 


153: 531,1035,1053,1350,1503,1530
. . 135: 1125,1152,1215,1251,1512,1521
. . 315: 1455,1545,1554
. . 351: 207,720
. . . . 27: 300
. . . . . . 3: 111,1011,1101,1110
. . . . . . 30: 1113,1131,1311
. . . . 72: 42,204,240,402,420
. . . . . . 24: 222
. . . . 270: 336,363,633
. . . . 702: 1548
. . . . . . 1458: 909,990
. . . . . . . . 99: 234,324,423
. . . . . . . . . . 243: 306,630
. . . . . . . . . . . . 36: 123,132,213,231,312,321,1023,1203
. . . . . . . . . . . . . . \ 1230,1302,1320
. . . . . . . . . . . . . . 1032: 828
. . . . . . . . . . . . . . . . 288: 246,264,426,462,642
. . . . . . . . . . . . . . . . . . 624: 1476
. . . . . . . . . . . . . . . . . . . . 1467: 1299
# . . . . . . . . . . . . . . . 882: 387,837,873,1359,1395,1539
. . . . . . . . . . . . . . . . . . \ 1593
. . . . . . . . . . . . . . . . . . 378: 237,273,327,723
. . . . . . . . . . . . . . . . . . . . 372: 1173,1317,1371
. . . . . . . . . . . . . . . . . . . . . . 1137: 1479,1497
. . . . . . . . . . . . . . . . . . . . 732: 1119,1191
. . . . . . . . . . . . . . . . . . 738: 291,912,921,1092,1209
. . . . . . . . . . . . . . . . . . . . \ 1290
. . . . . . . . . . . . . . . . . . . . 129: 144,414,1044,1404
. . . . . . . . . . . . . . . . . . . . . . \ 1440
. . . . . . . . . . . . . . . . . . . . . . 441: 1266
. . . . . . . . . . . . . . . . . . . . 192: 444
. . . . . . . . . . . . . . . . . . . . 219: 1116
. . . . . . . . . . . . . . . . . . . . . . 1161: 669,696,966
. . . . . . . . . . . . . . . . . . . . 1029: 777
. . . . . . . . . . . . . . . . . . 783: 339,393,933
. . . . . . . . . . . . 63: 1233,1323,1332
. . . . . . . . . . . . 360: 1227,1272
. . . . . . . . . . . . 603: 348,384,438,483,834,843
. . . . . . . . . . 342: 165,516,615,651,1056,1065,1506,1560
. . . . . . . . . . . . 156: 1344,1434,1443
. . . . . . . . . . . . 561: 1167,1176
. . . . . . . . . . 432: 606,660
. . . . . . . . . . . . 66: 114,411,1014,1041,1104,1140,1401
. . . . . . . . . . . . . . \ 1410
. . . . . . . . . . . . . . 141:
. . . . . . . . . . . . . . . . 225: 162,261,612,621,1062,1206
. . . . . . . . . . . . . . . . . . \ 1260
. . . . . . . . . . . . . . . . . . 126: 15,51,105,150,501,510
. . . . . . . . . . . . . . . . . . . . \ 1005,1050,1500
. . . . . . . . . . . . . . . . . . 216: 6,60,354,453,534,543
. . . . . . . . . . . . . . . . . . . . \ 600
. . . . . . . . . . . . . . . . . . . . 345: 117,171,711,1017
. . . . . . . . . . . . . . . . . . . . . . \ 1107,1170,1446,1464
. . . . . . . . . . . . . . . . . . . . . . 1071: 678,768,786,867
. . . . . . . . . . . . . . . . . . . . . . . . \ 876,1569,1596
. . . . . . . . . . . . . . . . . . . . . . . .687: 177,717,771,1077
. . . . . . . . . . . . . . . . . . . . 435: 1347,1374,1437,1473
. . . . . . . . . . . . . . . . . . 1026: 1188
. . . . . . . . . . . . . . . . 252: 1155,1236,1263,1326,1362
. . . . . . . . . . . . . . . . . . \ 1515,1551
. . . . . . . . . . . . . . . . 522: 1128,1182,1218,1281
. . . . . . 1485: 399,939,993
. . . . . . 1584: 789,798,879,897,978,987,1599
 . . 513: 108,810,1008
. . . . 18: 1122,1212,1221
. . . . 81: 333,1224,1422
. . . . . . 1242: 819,891,1098
. . . . . . . . 189: 45,450,504
. . . . . . . . . . 54: 33,303,330
. . . . . . . . . . 405: 456,465,546,564,654
. . . . . . . . . . . . 645: 285,582,825,852
. . . . . . . . . . . . . . 258: 255,525,552
. . . . . . . . . . . . . . 528: 228,822
. . . . . . . . . . . . . . . . 282: 1146,1164,1416,1461
. . . . . . . . . . 540: 138,183,318,381,813,831,1038,1083,1308
. . . . . . . . . . . . \ 1380
. . . . . . . . 198: 1245,1254,1425,1452,1524,1542
. . . . . . . . 918: 549
. . . . . . . . . . 459: 366,636,663
. . . . . . . . . . 495: 357,537,573,735,753
. . . . . . . . . . . . 375: 555
. . . . . . . . . . 594: 1557,1575
. . . . . . . . . . 945: 69,96,609,690,906,960
. . . . . . . . . . 954: 1296
. . . . . . . . . . . . 1269: 1389,1398
. . . . . . . . 981: 1578,1587
. . . . . . . . 1089: 1488
. . . . 180: 1335,1353,1533
. . . . 801: 249,294,429,492,924,942
. . . . 1080: 279,297,927
. . . . . . 729: 90,168,186,618,681,816,861,900,1068,1086
. . . . . . . . 9: 12,21,102,120,201,210,1002,1020,1200
. . . . . . 792: 486,846
. . . . . . . . 468: 57,75,507,570,705
. . . . . . . . . . 750: 747,774
. . . . . . . . . . . . 477: 1257,1275,1527,1572
. . . . . . . . 648: 666
. . . . . . . . 684: 657,675
. . . . . . . . . . 567: 267,276,627,672,726,1338,1383
. . . . . . . . . . . . 762:
. . . . . . . . . . . . . . 558: 1566
. . . . . . . . . . . . . . 585: 1248,1284,1428,1482
. . . . . . . . . . . . . . 855: 78,87,159,195,519,591,708,780
. . . . . . . . . . . . . . . . \ 807,870,915,951,1059,1095,1509
. . . . . . . . . . . . . . . . \ 1590
. . . . . . . . . . 576: 48,84,480,804,840
. . . . . . . . . . . . 408: 147,174,417,741,1047,1407,1470
. . . . . . . . . . . . . . 471: 447,474,744
. . . . . . . . . . . . . . 714: 1377
. . . . . . . . . . . . . . 1074: 1179
. . . . . . . . . . . . . . . . 1197: 579,597,759,957,975
. . . . . . . . . . . . . . . . . . 795: 1194,1419,1491
. . . . . . . . . . . . . . . . . . . . 1149: 588,885
. . . . . . . . . . . . . . . . . . . . . . 858: 1449,1494
. . . . . . . . . . 756: 39,309,390,903,930,1368,1386
. . . . . . . . . . . . 93: 1134,1143,1314,1341,1413,1431
. . . . . . . . . . 765: 1239,1293,1329,1392
. . . . . . . . 864: 1278,1287
. . . . . . 972: 396,693,936,963
. . . . . . . . 369: 1356,1365,1563
. . . . . . . . . . 1536: 888
. . . . . . . . 639: 1158,1185,1518,1581
. . 1305: 489,498,849,894,948,984