Variations of Erase and Triple

In my previous post, I looked at the traditional "erase and triple" protocol which eliminates all redundant digits in a number or, if there are none, triples the number. For example, consider 112:

112 --> 2 --> 6 --> 18 --> 54 --> 4 --> 12 --> 1 --> 3 --> 0

In that post, I looked at record heights and record lengths reached by numbers when this protocol is repeatedly applied. In this post, I'll propose a variation of the protocol. Instead of the erasure of repeated digits, I'll remove any prime digits (2, 3, 5 and 7). If there are no prime digits, then the number will be tripled. Let's use 112 again as an example:

112 --> 11 --> 33 --> 0

Let's try 66 with no prime digits and see what happens to this number:

66 --> 198 --> 594 --> 94 --> 282 --> 8 --> 24 --> 4 -->  12 --> 1 --> 3 --> 0

A pattern seems to be emerging. Let's test out what the record trajectory lengths are for the first 500,000 numbers (permalink):

Erase prime digits or multiply by 3: record breakers

 1 --> 3 --> [1, 3, 0]

4 --> 5 --> [4, 12, 1, 3, 0]

6 --> 8 --> [6, 18, 54, 4, 12, 1, 3, 0]

16 --> 9 --> [16, 48, 144, 432, 4, 12, 1, 3, 0]

66 --> 12 --> [66, 198, 594, 94, 282, 8, 24, 4, 12, 1, 3, 0]

166 --> 13 --> [166, 498, 1494, 4482, 448, 1344, 144, 432, 4, 12, 1, 3, 0]

666 --> 16 --> [666, 1998, 5994, 994, 2982, 98, 294, 94, 282, 8, 24, 4, 12, 1, 3, 0]

1666 --> 17 --> [1666, 4998, 14994, 44982, 4498, 13494, 1494, 4482, 448, 1344, 144, 432, 4, 12, 1, 3, 0]

6666 --> 20 --> [6666, 19998, 59994, 9994, 29982, 998, 2994, 994, 2982, 98, 294, 94, 282, 8, 24, 4, 12, 1, 3, 0]

16666 --> 21 --> [16666, 49998, 149994, 449982, 44998, 134994, 14994, 44982, 4498, 13494, 1494, 4482, 448, 1344, 144, 432, 4, 12, 1, 3, 0]

66666 --> 24 --> [66666, 199998, 599994, 99994, 299982, 9998, 29994, 9994, 29982, 998, 2994, 994, 2982, 98, 294, 94, 282, 8, 24, 4, 12, 1, 3, 0]

166666 --> 25 --> [166666, 499998, 1499994, 4499982, 449998, 1349994, 149994, 449982, 44998, 134994, 14994, 44982, 4498, 13494, 1494, 4482, 448, 1344, 144, 432, 4, 12, 1, 3, 0]

The numbers are shown first, followed by length of their trajectories and then the trajectories themselves. It can be seen that 66 is one of record breaking numbers. Apart from 4, all the numbers contain only 1 and 6 as digits. Figure 1 shows the graph of the trajectory of 166666. 

Figure 1: permalink

As for record heights, it's clear that these will increase as the numbers get larger because any number not containing any prime digits will immediately be trebled. After 166666 with a trajectory length of 25, there is big gap to the next record length which is attained by 666666 with a length of 28. The trajectory is:

666666, 1999998, 5999994, 999994, 2999982, 99998, 299994, 99994, 299982, 9998, 29994, 9994, 29982, 998, 2994, 994, 2982, 98, 294, 94, 282, 8, 24, 4, 12, 1, 3, 0

The trajectory is shown in Figure 2 and is very similar to that of 166666 shown in Figure 1.

Figure 2: permalink

All the numbers that I've checked end in 0 mostly via 3 but not always. For example the trajectory of 1948 is:

1948 --> 5844 --> 844 --> 2532 --> 0

The end result always seems to be zero. The next thing to logically explore is what happens when we change the multiplier of the number from 3 to some other number. Let's start with 2 and work our way up from 4 to 11, showing in every case the record breakers up to 500,000.

Erase prime digits or multiply by 2: record breakers

1 --> 3 --> [1, 2, 0]

4 --> 5 --> [4, 8, 16, 32, 0]

9 --> 8 --> [9, 18, 36, 6, 12, 1, 2, 0]

29 --> 9 --> [29, 9, 18, 36, 6, 12, 1, 2, 0]

46 --> 10 --> [46, 92, 9, 18, 36, 6, 12, 1, 2, 0]

49 --> 12 --> [49, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

149 --> 13 --> [149, 298, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

498 --> 16 --> [498, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

999 --> 18 --> [999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

2999 --> 19 --> [2999, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

4996 --> 20 --> [4996, 9992, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

4999 --> 22 --> [4999, 9998, 19996, 39992, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

14999 --> 23 --> [14999, 29998, 9998, 19996, 39992, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

49998 --> 26 --> [49998, 99996, 199992, 19999, 39998, 9998, 19996, 39992, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

99999 --> 28 --> [99999, 199998, 399996, 99996, 199992, 19999, 39998, 9998, 19996, 39992, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

299999 --> 29 --> [299999, 99999, 199998, 399996, 99996, 199992, 19999, 39998, 9998, 19996, 39992, 999, 1998, 3996, 996, 1992, 199, 398, 98, 196, 392, 9, 18, 36, 6, 12, 1, 2, 0]

Erase prime digits or multiply by 4: record breakers

1 --> 8 --> [1, 4, 16, 64, 256, 6, 24, 4]

9 --> 9 --> [9, 36, 6, 24, 4, 16, 64, 256, 6] 

11 --> 10 --> [11, 44, 176, 16, 64, 256, 6, 24, 4, 16]

41 --> 12 --> [41, 164, 656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

99 --> 13 --> [99, 396, 96, 384, 84, 336, 6, 24, 4, 16, 64, 256, 6]

101 --> 16 --> [101, 404, 1616, 6464, 25856, 86, 344, 44, 176, 16, 64, 256, 6, 24, 4, 16]

104 --> 17 --> [104, 416, 1664, 6656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

604 --> 18 --> [604, 2416, 416, 1664, 6656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

901 --> 20 --> [901, 3604, 604, 2416, 416, 1664, 6656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

2901 --> 21 --> [2901, 901, 3604, 604, 2416, 416, 1664, 6656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

10401 --> 23 --> [10401, 41604, 166416, 665664, 66664, 266656, 6666, 26664, 6664, 26656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

41666 --> 24 --> [41666, 166664, 666656, 66666, 266664, 66664, 266656, 6666, 26664, 6664, 26656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

90104 --> 26 --> [90104, 360416, 60416, 241664, 41664, 166656, 16666, 66664, 266656, 6666, 26664, 6664, 26656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64] 

104166 --> 27 --> [104166, 416664, 1666656, 166666, 666664, 2666656, 66666, 266664, 66664, 266656, 6666, 26664, 6664, 26656, 666, 2664, 664, 2656, 66, 264, 64, 256, 6, 24, 4, 16, 64]

Erase prime digits or multiply by 5: record breakers

 1 --> 3 --> [1, 5, 0]

8 --> 4 --> [8, 40, 200, 0] 

9 --> 5 --> [9, 45, 4, 20, 0] 

18 --> 6 --> [18, 90, 450, 40, 200, 0] 

19 --> 7 --> [19, 95, 9, 45, 4, 20, 0

129 --> 8 --> [129, 19, 95, 9, 45, 4, 20, 0] 

199 --> 9 --> [199, 995, 99, 495, 49, 245, 4, 20, 0] 

1299 --> 10 --> [1299, 199, 995, 99, 495, 49, 245, 4, 20, 0] 

1999 --> 11 --> [1999, 9995, 999, 4995, 499, 2495, 49, 245, 4, 20, 0] 

12999 --> 12 --> [12999, 1999, 9995, 999, 4995, 499, 2495, 49, 245, 4, 20, 0] 

19999 --> 13 --> [19999, 99995, 9999, 49995, 4999, 24995, 499, 2495, 49, 245, 4, 20, 0] 

129999 --> 14 --> [129999, 19999, 99995, 9999, 49995, 4999, 24995, 499, 2495, 49, 245, 4, 20, 0]

199999 --> 15 --> [199999, 999995, 99999, 499995, 49999, 249995, 4999, 24995, 499, 2495, 49, 245, 4, 20, 0]

Erase prime digits or multiply by 6: record breakers

1 --> 4 --> [1, 6, 36, 6]

8 --> 6 --> [8, 48, 288, 88, 528, 8]

11 --> 8 --> [11, 66, 396, 96, 576, 6, 36, 6]

18 --> 12 --> [18, 108, 648, 3888, 888, 5328, 8, 48, 288, 88, 528, 8]

68 --> 15 --> [68, 408, 2448, 448, 2688, 688, 4128, 418, 2508, 8, 48, 288, 88, 528, 8]

69 --> 16 --> [69, 414, 2484, 484, 2904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

144 --> 21 --> [144, 864, 5184, 184, 1104, 6624, 664, 3984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

194 --> 26 --> [194, 1164, 6984, 41904, 251424, 144, 864, 5184, 184, 1104, 6624, 664, 3984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

1199 --> 28 --> [1199, 7194, 194, 1164, 6984, 41904, 251424, 144, 864, 5184, 184, 1104, 6624, 664, 3984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

1699 --> 35 --> [1699, 10194, 61164, 366984, 66984, 401904, 2411424, 41144, 246864, 46864, 281184, 81184, 487104, 48104, 288624, 8864, 53184, 184, 1104, 6624, 664, 3984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

12699 --> 36 --> [12699, 1699, 10194, 61164, 366984, 66984, 401904, 2411424, 41144, 246864, 46864, 281184, 81184, 487104, 48104, 288624, 8864, 53184, 184, 1104, 6624, 664, 3984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

101199 --> 37 --> [101199, 607194, 60194, 361164, 61164, 366984, 66984, 401904, 2411424, 41144, 246864, 46864, 281184, 81184, 487104, 48104, 288624, 8864, 53184, 184, 1104, 6624, 664, 3984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4]

184164 --> 38 --> [184164, 1104984, 6629904, 669904, 4019424, 401944, 2411664, 411664, 2469984, 469984, 2819904, 819904, 4919424, 491944, 2951664, 91664, 549984, 49984, 299904, 99904, 599424, 9944, 59664, 9664, 57984, 984, 5904, 904, 5424, 44, 264, 64, 384, 84, 504, 4, 24, 4] 

Erase prime digits or multiply by 7: record breakers

1 --> 3 --> [1, 7, 0]

4 --> 7 --> [4, 28, 8, 56, 6, 42, 4]

9 --> 9 --> [9, 63, 6, 42, 4, 28, 8, 56, 6]

14 --> 13 --> [14, 98, 686, 4802, 480, 3360, 60, 420, 40, 280, 80, 560, 60]

84 --> 14 --> [84, 588, 88, 616, 4312, 41, 287, 8, 56, 6, 42, 4, 28, 8]

184 --> 15 --> [184, 1288, 188, 1316, 116, 812, 81, 567, 6, 42, 4, 28, 8, 56, 6]

406 --> 16 --> [406, 2842, 84, 588, 88, 616, 4312, 41, 287, 8, 56, 6, 42, 4, 28, 8]

988 --> 17 --> [988, 6916, 48412, 4841, 33887, 88, 616, 4312, 41, 287, 8, 56, 6, 42, 4, 28, 8]

1404 --> 19 --> [1404, 9828, 988, 6916, 48412, 4841, 33887, 88, 616, 4312, 41, 287, 8, 56, 6, 42, 4, 28, 8]

4164 --> 21 --> [4164, 29148, 9148, 64036, 6406, 44842, 4484, 31388, 188, 1316, 116, 812, 81, 567, 6, 42, 4, 28, 8, 56, 6]

9964 --> 22 --> [9964, 69748, 6948, 48636, 4866, 34062, 406, 2842, 84, 588, 88, 616, 4312, 41, 287, 8, 56, 6, 42, 4, 28, 8]

29964 --> 23 --> [29964, 9964, 69748, 6948, 48636, 4866, 34062, 406, 2842, 84, 588, 88, 616, 4312, 41, 287, 8, 56, 6, 42, 4, 28, 8]

99144 --> 26 --> [99144, 694008, 4858056, 48806, 341642, 4164, 29148, 9148, 64036, 6406, 44842, 4484, 31388, 188, 1316, 116, 812, 81, 567, 6, 42, 4, 28, 8, 56, 6]

299144 --> 27 --> [299144, 99144, 694008, 4858056, 48806, 341642, 4164, 29148, 9148, 64036, 6406, 44842, 4484, 31388, 188, 1316, 116, 812, 81, 567, 6, 42, 4, 28, 8, 56, 6] 

Erase prime digits or multiply by 8: record breakers

1 --> 5 --> [1, 8, 64, 512, 1]

6 --> 6 --> [6, 48, 384, 84, 672, 6]

14 --> 8 --> [14, 112, 11, 88, 704, 4, 32, 0]

16 --> 11 --> [16, 128, 18, 144, 1152, 11, 88, 704, 4, 32, 0]

81 --> 13 --> [81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0]

281 --> 14 --> [281, 81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0]

1019 --> 15 --> [1019, 8152, 81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0]

6018 --> 16 --> [6018, 48144, 385152, 81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0]

8001 --> 18 --> [8001, 64008, 512064, 1064, 8512, 81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0]

16101 --> 19 --> [16101, 128808, 18808, 150464, 10464, 83712, 81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0]

100019 --> 20 --> [100019, 800152, 8001, 64008, 512064, 1064, 8512, 81, 648, 5184, 184, 1472, 14, 112, 11, 88, 704, 4, 32, 0] 

Erase prime digits or multiply by 9: record breakers

1 --> 5 --> [1, 9, 81, 729, 9]

11 --> 6 --> [11, 99, 891, 8019, 72171, 11]

14 --> 25 --> [14, 126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

124 --> 26 --> [124, 14, 126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

198 --> 27 --> [198, 1782, 18, 162, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

666 --> 30 --> [666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

1184 --> 32 --> [1184, 10656, 1066, 9594, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

6996 --> 34 --> [6996, 62964, 6964, 62676, 666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

26996 --> 35 --> [26996, 6996, 62964, 6964, 62676, 666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

40444 --> 36 --> [40444, 363996, 6996, 62964, 6964, 62676, 666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

44916 --> 38 --> [44916, 404244, 40444, 363996, 6996, 62964, 6964, 62676, 666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

49904 --> 40 --> [49904, 449136, 44916, 404244, 40444, 363996, 6996, 62964, 6964, 62676, 666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6]

249904 --> 41 --> [249904, 49904, 449136, 44916, 404244, 40444, 363996, 6996, 62964, 6964, 62676, 666, 5994, 994, 8946, 80514, 8014, 72126, 16, 144, 1296, 196, 1764, 164, 1476, 146, 1314, 114, 1026, 106, 954, 94, 846, 7614, 614, 5526, 6, 54, 4, 36, 6] 

Erase prime digits or multiply by 11: record breakers

1 --> 4 --> [1, 11, 121, 11]

4 --> 5 --> [4, 44, 484, 5324, 4]

8 --> 11 --> [8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

19 --> 12 --> [19, 209, 9, 99, 1089, 11979, 1199, 13189, 1189, 13079, 109, 1199]

48 --> 13 --> [48, 528, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

68 --> 15 --> [68, 748, 48, 528, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

81 --> 20 --> [81, 891, 9801, 107811, 10811, 118921, 11891, 130801, 10801, 118811, 1306921, 10691, 117601, 11601, 127611, 1611, 17721, 11, 121, 11]

98 --> 23 --> [98, 1078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

298 --> 24 --> [298, 98, 1078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

848 --> 25 --> [848, 9328, 98, 1078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

2848 --> 26 --> [2848, 848, 9328, 98, 1078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

4198 --> 27 --> [4198, 46178, 4618, 50798, 98, 1078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

8018 --> 32 --> [8018, 88198, 970178, 9018, 99198, 1091178, 109118, 1200298, 10098, 111078, 11108, 122188, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

14918 --> 35 --> [14918, 164098, 1805078, 18008, 198088, 2178968, 18968, 208648, 8648, 95128, 918, 10098, 111078, 11108, 122188, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

80998 --> 36 --> [80998, 890978, 89098, 980078, 98008, 1078088, 108088, 1188968, 13078648, 108648, 1195128, 11918, 131098, 11098, 122078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

81408 --> 37 --> [81408, 895488, 89488, 984368, 98468, 1083148, 108148, 1189628, 118968, 1308648, 108648, 1195128, 11918, 131098, 11098, 122078, 108, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

84088 --> 41 --> [84088, 924968, 94968, 1044648, 11491128, 1149118, 12640298, 164098, 1805078, 18008, 198088, 2178968, 18968, 208648, 8648, 95128, 918, 10098, 111078, 11108, 122188, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198]

284088 --> 42 --> [284088, 84088, 924968, 94968, 1044648, 11491128, 1149118, 12640298, 164098, 1805078, 18008, 198088, 2178968, 18968, 208648, 8648, 95128, 918, 10098, 111078, 11108, 122188, 1188, 13068, 1068, 11748, 1148, 12628, 168, 1848, 20328, 8, 88, 968, 10648, 117128, 1118, 12298, 198, 2178, 18, 198] 

Thus all these "variants" have trajectories that either terminate in 0 or they loop. For example, 284088 has a trajectory of length 42 and it ends in a loop (198 --> 2178 --> 18 --> 198). Figure 3 shows the trajectory of 284088:

Figure 4: permalink

The original erase or triple and the variants analysed in this post all have one thing in common. The starting number is decreased in some way according to a certain rule e.g. remove duplicate digits or remove prime digits. If the rule cannot be applied, then the number is increased in some way e.g. it is tripled. I briefly tested an alternative rule to tripling, specifically squaring the number, but found that the resultant numbers quickly grew without bound.

It would be nice to find a balance between the rule that decreases the number and the rule that increases it, so that sometimes the trajectory terminates while in other cases it does not. Clearly with the erase or triple protocol, the resulting number chain quickly terminates because of a loop or a number being reached that causes a complete or almost complete collapse because all or most of its digits are repeated or all or most of its digits are prime.

Even multiplying by a relatively large number eventually leads to a termination. For example, multiply by 99 when there are no prime digits and use 1 as the example. Here is the trajectory that consists of 238 steps and ends in a loop:

1, 99, 9801, 970299, 9099, 900801, 89179299, 891999, 88307901, 880901, 87209199, 809199, 80110701, 8011001, 793089099, 9089099, 899820801, 89980801, 8908099299, 890809999, 88190189901, 8730828800199, 8088800199, 800791219701, 800911901, 79290278199, 9908199, 980911701, 98091101, 9711018999, 911018999, 90190880901, 8928897209199, 8988909199, 889902010701, 8899001001, 881001099099, 87219108810801, 819108810801, 81091772269299, 810916999, 80280782901, 80808901, 8000081199, 792008038701, 90080801, 8917999299, 89199999, 8830799901, 88099901, 8721890199, 81890199, 8107129701, 8101901, 802088199, 80088199, 7928731701, 98101, 9711999, 911999, 90287901, 908901, 89981199, 8908138701, 89081801, 8819098299, 881909899, 87309080001, 809080001, 80098920099, 8009890099, 792979119801, 999119801, 98912860299, 989186099, 97929423801, 9994801, 989485299, 9894899, 979595001, 999001, 98901099, 9791208801, 99108801, 9811771299, 9811199, 971308701, 910801, 90169299, 9016999, 892682901, 8968901, 887921199, 8891199, 880228701, 880801, 87199299, 819999, 81179901, 8119901, 803870199, 8080199, 799939701, 999901, 98990199, 9800029701, 98000901, 9702089199, 90089199, 8918830701, 89188001, 8829612099, 88961099, 8807148801, 880148801, 87134731299, 814199, 80605701, 806001, 79794099, 994099, 98415801, 9841801, 974338299, 94899, 9395001, 99001, 9801099, 970308801, 9008801, 891871299, 8918199, 882901701, 8890101, 880119999, 87131879901, 81189901, 8037800199, 80800199, 7999219701, 9991901, 989198199, 97930621701, 9906101, 980703999, 9800999, 970298901, 9098901, 900791199, 90091199, 8919028701, 89190801, 8829889299, 88988999, 8809910901, 872181179199, 818119199, 80993800701, 809980001, 80188020099, 8018800099, 793861209801, 986109801, 97624870299, 9648099, 955161801, 9161801, 907018299, 9001899, 891188001, 88227612099, 8861099, 877248801, 848801, 84031299, 840199, 83179701, 81901, 8108199, 802711701, 801101, 79308999, 908999, 89990901, 8909099199, 882000820701, 880008001, 87120792099, 8109099, 802800801, 80800801, 7999279299, 999999, 98999901, 9800990199, 970298029701, 90980901, 9007109199, 900109199, 89110810701, 8911081001, 882197019099, 8819019099, 873082890801, 808890801, 80080189299, 8008018999, 792793880901, 99880901, 9888209199, 988809199, 97892110701, 98911001, 9792189099, 99189099, 9819720801, 98190801, 9720889299, 9088999, 899810901, 89081279199, 890819199, 88191100701, 8819110001, 873091890099, 8091890099, 801097119801, 80109119801, 7930802860299, 908086099, 89900523801, 89900801, 8900179299, 89001999, 8811197901, 881119901, 87230870199, 8080199

These ruminations are fertile grounds for further posts involving new rules.