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.

Erase or Triple Protocol

There's always something new to discover under the mathematical sun and today I encountered an interesting protocol that can be applied to numbers. It works as follows:

The "Erase or triple" protocol describes how to transform an integer \(K\) into an integer \(L\): if \(K\) has 2 or more identical digits, erase them to get \(L\) (1201331 becomes 20); if \(K\) has no duplicate digits, triple \(K\) to get \(L\) (20 becomes 60). Some integers disappear immediately (like 11, 2002 or 1919188), other enter into a loop if you apply this protocol to the successive results. Link.

My diurnal age today was 26784, a number that is a member of OEIS A300150: "erase or triple": list of the successive integers that produce the next "altitude" record. The initial numbers and their associated "altitude" records are as follows (the numbers are shown first in bold and records second - permalink):

(1, 17010), (10, 65610), (23, 121743), (176, 1154736), (1760, 1283040), (2183, 1591407), (2640, 5773680), (23976, 5826168), (24056, 5845608), (26784, 6508512), (29087, 7068141), (29701, 7217343), (30715, 7463745), (31456, 7643808), (32145, 7811235)

Thus the numbers associated with these maximum values are:

1  10  23  176  1760  2183  2640  23976  24056  26784  29087  29701  30715  31456  32145  

 Let's take 26784 as an example. It's trajectory is as follows:

26784, 80352, 241056, 723168, 2169504, 6508512, 60812, 182436, 547308, 1641924, 692, 2076, 6228, 68, 204, 612, 1836, 5508, 8, 24, 72, 216, 648, 1944, 19, 57, 171, 7, 21, 63, 189, 567, 1701, 70, 210, 630, 1890, 5670, 17010, 7

Note that trajectory enters a loop once 7 is reached for the second time. The trajectory has a length of 39 steps. It's graph is shown in Figure 1:

Figure 1: permalink

176, from the above list of record breakers, is an example of a number that eventually reaches 0. It's trajectory is as follows:

176, 528, 1584, 4752, 14256, 42768, 128304, 384912, 1154736, 54736, 164208, 492624, 96, 288, 2, 6, 18, 54, 162, 486, 1458, 4374, 37, 111, 0

The graph of its trajectory is shown in Figure 2 and consists of 25 steps: 

Figure 2: permalink

23, from the above list of record breakers, is an example of a number that ends in an 89 loop. It's trajectory of length 21 steps is as follows:

23, 69, 207, 621, 1863, 5589, 89, 267, 801, 2403, 7209, 21627, 167, 501, 1503, 4509, 13527, 40581, 121743, 2743, 8229, 89

The graph of its trajectory is shown in Figure 3:

Figure 3: permalink

29701, from the above list of record breakers, is an example of a number that ends in a 5 loop. It's trajectory, of length 22 steps, is as follows:

29701, 89103, 267309, 801927, 2405781, 7217343, 214, 642, 1926, 5778, 58, 174, 522, 5, 15, 45, 135, 405, 1215, 25, 75, 225, 5

The graph of its trajectory is shown in Figure 4:

Figure 4: permalink

For any number, there are only four possible end results for its trajectory: either it reaches 0 or it enters a 5, 7 or 89 loop. Returning to OEIS A300150: "erase or triple": list of the successive integers that produce the next "altitude" record. The sequence is finite and has 628 terms, with a(628) = 3291768054 (pandigital); a(628) reaches the maximum possible "altitude" 29625912486.

When dealing with trajectories, we are interested in the length of the trajectories as well as the maxima and so a reasonable question to ask is what numbers produce trajectories of record length? It turns out that these are the records up to 40,000 with numbers first in bold and trajectory lengths following (permalink):

[(1, 29), (16, 32), (26, 35), (56, 37), (134, 39), (218, 41), (241, 45), (871, 46), (8059, 47), (14957, 48)]

Thus the numbers associated with the record trajectory lengths are:

1  16  26  56  134  218   241  871   8059  14957 

 The trajectory of 14957, with a trajectory length of 48, is as follows:

14957, 44871, 871, 2613, 7839, 23517, 70551, 701, 2103, 6309, 18927, 56781, 170343, 1704, 5112, 52, 156, 468, 1404, 10, 30, 90, 270, 810, 2430, 7290, 21870, 65610, 510, 1530, 4590, 13770, 130, 390, 1170, 70, 210, 630, 1890, 5670, 17010, 7, 21, 63, 189, 567, 1701, 70

This 70 loop is actually a part of the 7 loop as can be seen below:

7, 21, 63, 189, 567, 1701, 70, 210, 630, 1890, 5670, 17010, 7 

The graph of its trajectory is shown in Figure 5:

Figure 5: permalink

This "erase or triple" protocol could be generalised so if the digits of a number satisfy a certain criterion then they are erased to form a new number or, if the criterion is not met, the number is modified in some way. For example, suppose the number contains prime digits (2, 3, 5 or 7). If it does, then these digits are erased. If the number does not contain any prime digits, then the number is squared and 1 is added.

Let's use 14857 as a test number. It contains the prime digits 5 and 7 so these are erased to leave 148. This number contains no prime digits so it becomes 148 x 148 + 1 = 21905. We erase the 2 and the 5 to get 190 which becomes 190 x 190 + 1 = 36101 and so on. I could continue but the trajectory under this new protocol is best dealt with by creating an appropriate algorithm (permalink).

Using the algorithm, the trajectory turns out to have a length of 7 and is:

14857, 148, 21905, 190, 36101, 6101, 37222202, 0

Once 37222202 is reached and all the prime digits are erased, we are left with 0. This seems to be the fate of many numbers but not all. For example, while 6, 66 and 666 all end in 0, 6666 increases rapidly without bound. It might be better to double the number and add 1 rather than squaring it and adding 1. However, I'm digressing. This new protocol and variations thereof could serve as the basis for a future post but that's enough for this post.

In closing, I'll just observe that protocols like these, where we are manipulating the digits of the number in some way, fall into the realm of recreational mathematics rather than serious mathematics. Not only are they specific to the number base 10 but they also ignore, in the first step, the place value of the digits and acknowledge only the face value. Nonetheless, it's fun to explore the resultant trajectories when the different protocols are applied.

Every Number Is Interesting

It's been quite some time since I last posted. My temporary relocation to the wilds of Sumatra has hampered my blogging opportunities but I wanted to make at least one post before July is over. It's July 31st now but on July 29th I turned 26780 days old. Initially, I could find little of interest about this number but I knew this was only because I wasn't looking hard enough. 

Once I did I soon realised that 26780 is special in at least one way, namely none of its digits repeat. This led me to ask how many such five digit numbers have this property? The five digit numbers range from 10000 to 99999 and there are thus 89999 of them. If there are to be five digits and none of them can repeat then the first, or leftmost digit, cannot be zero so there are 9 possibilities. For the second digit there are also 9 possibilities, followed by 8, 7 and 6 for the remaining positions, giving at total of 9 x 9 x 8 x 7 x 6 = 27216 numbers. This represents 30.24% of the range. 

Thus 26780 is special but not that special because about three out of every ten five digit numbers have this same property. Examining the number further, I noticed that the digit sum of 26780 is 23 and thus it has a digital root of five. Thus it's a five digit number with a digital root of five. Numbers with these two properties are ten times as rare. In fact there are 3024 of them giving a percentage of 3.36%. This makes the number a lot more special.

On still further investigation, I noticed that 26780 has five prime factors, counting multiplicity, because it factorises to 2 x 2 x 5 x 13 x 103. There are only  253 such numbers out of the 89999 and this represents 0.0028%, making the number something of a rara avis. Here is the list of such numbers:

10256, 10625, 10832, 12056, 12650, 13064, 13208, 13496, 14036, 14360, 14576, 14630, 14756, 14792, 15260, 15368, 15620, 15728, 15980, 16304, 16340, 16952, 17204, 17240, 17384, 17420, 17456, 17528, 17960, 18536, 18590, 19256, 19472, 19580, 19652, 19832, 20984, 23180, 23450, 23540, 23576, 23864, 24368, 25016, 25340, 25916, 25970, 26780, 27140, 27536, 27860, 27896, 27950, 28175, 28490, 28760, 28976, 29768, 29876, 30416, 30758, 30875, 31064, 31208, 31496, 31568, 31784, 31820, 31892, 31928, 32450, 32504, 34250, 34520, 34790, 34916, 34952, 35096, 35240, 36140, 36248, 36752, 37625, 37940, 38012, 38120, 38570, 39056, 39128, 39416, 39650, 39704, 39875, 40136, 40172, 40568, 40712, 41756, 41936, 42980, 43016, 43250, 43610, 45032, 45392, 46508, 46580, 46832, 47012, 47138, 47192, 47912, 48650, 48920, 49028, 49352, 49532, 49820, 50468, 50648, 50792, 51260, 51620, 51980, 52016, 52430, 53168, 53690, 53924, 54230, 54392, 54608, 54680, 54860, 56012, 56210, 56408, 56912, 56984, 57128, 57380, 58190, 58460, 58712, 59180, 59216, 59720, 59864, 60125, 60152, 60872, 61304, 62150, 62348, 62510, 62780, 63752, 63824, 64130, 65048, 65192, 65219, 65480, 67208, 67298, 67928, 68072, 68324, 68450, 68540, 69125, 69350, 69584, 70196, 71825, 71852, 72590, 72968, 73508, 73580, 73625, 73850, 74192, 75164, 75416, 75803, 76280, 76532, 76820, 79160, 79250, 79268, 79304, 79430, 80132, 80276, 80456, 80465, 81032, 81356, 81392, 81752, 81950, 82076, 82760, 82904, 83912, 84632, 85316, 85640, 85910, 86504, 86792, 87125, 87620, 89420, 89456, 89672, 89726, 90248, 90356, 90428, 91364, 91472, 91580, 91832, 91850, 92876, 93128, 93560, 93740, 94280, 94316, 94352, 94820, 94856, 95108, 95432, 95612, 95720, 95810, 95864, 96278, 96350, 96530, 96728, 97016, 97250, 97340, 97682, 98150, 98456, 98735

Such numbers might well find a place in the Online Encyclopedia of Integer Sequence (OEIS) but I'll not be proposing them as a sequence given that I don't contribute anymore. The sequence can be stated as follows:

Five digit numbers in which no digit is repeated, the digital root is 5 and there are five prime factors (not necessarily distinct)

The reasons I've given for not interacting with the OEIS are stated in an earlier post. The point of this post is to show that every number has interesting properties if you look hard enough.

ADDENDUM: October 19th 2022

I had to include the following that I discovered here as it's relevant to the content of this post. 

Folklore tells us that there are no uninteresting natural numbers. The argument hinges on the following observation: Every subset of the natural numbers is either empty, or has a smallest element. The argument usually goes something like this. If there would be any uninteresting natural numbers, the set U of all these uninteresting natural numbers would have a smallest element, say u ∈ U. But u in itself has a very remarkable property. u is the smallest uninteresting natural number, which is very interesting indeed. So U, the set of all the uninteresting natural numbers, can not have a smallest element, therefore U must be empty. In other words, all natural numbers are interesting.

Multiplicative Persistence and Multiplicative Digital Root

My diurnal age today (26748) has the property that it has a multiplicative persistence of 6. This qualifies it for membership in OEIS A199996:

A199996

Composite numbers whose multiplicative persistence is 6.

The initial members of the sequence are:

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487, 26748, 26784, 26847, 26874, 27388 ...

To quote from Wikipedia:

In number theory, the multiplicative digital root of a natural number \(n\) in a given number base \(b\) is found by multiplying the digits of \(n\) together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of \(n\). Multiplicative digital roots are the multiplicative equivalent of digital roots.

The number of iterations required to reach the multiplicative digital root is termed the multiplicative persistence. It is conjectured that there is no number with a multiplicative persistence greater than 11. The smallest numbers with multiplicative persistence of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 are:

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899

These numbers constitute OEIS A003001 in the OEIS (see permalink for an algorithm that will generate the members of this sequence up to one million). 

Here is permalink to an algorithm that will calculate multiplicative persistence and multiplicative digital roots for a range of numbers (both composite and prime). The algorithm is easily modified to search for a specific multiplicative persistence or multiplicative digital root. Primes can be excluded by simply adding that condition to the relevant section of the code.

ADDENDUM

On October 4th 2022, my diurnal age was 26847, a permutation of the digits of 26748 and thus also having a multiplicative persistence of 6. In between these two diurnal ages, I passed 26784 days, another permutation, and 26874 is still to come. Altogether there are 120 permutations of the digits 2, 4, 6, 7 and 8. 

It's interesting to look at a breakdown of the percentages of numbers with various multiplicative persistences. Up to one million, the breakdown is:

7    0.245%

6    0.449%

5    2.47%

4    6.68%

3    12.4%

2    37.5%

1    40.3%

In this range there are no numbers with a multiplicative persistence of 8. The first such number is 2,677,889. 

Arithmetic Numbers

I was familiar with the concept of an arithmetic derivative but today I was reminded of the concept of an arithmetic number. I encounter the term almost daily when referring to the website Numbers Aplenty but hadn't paid it much attention. Here is a definition from Wikipedia:

In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 3:$$ \frac{1+2+3+6}{4}=3$$Phrasing it more rigorously, we could say that a number \(n\) is arithmetic if the number of divisors \(d(n)\) or \( \sigma_0(n) \) divides the sum of divisors \( \sigma_1(n) \).

This ratio is not be confused with the ratio that I was investigating in my previous post on Barely Abundant Numbers. In that case, the ratio was \( \sigma(n) \) over \(n\) which determines whether a number is deficient, perfect or abundant. If the ratio is less than 2, the number is deficient. If it's greater than 2, the number is abundant. We know 6 is a perfect number because the ratio equals 2:$$ \frac{1+2+3+6}{6}=2$$The term was brought to my attention by one of the properties of the number associated with my diurnal age of 26744. The property qualifies the number for membership in OEIS A107924 as the 119th member.

A107924

Even numbers \(n\) such that \(n^2\) is an arithmetic number.

296, 536, 632, 872, 1208, 1304, 1544, 2072, 2216, 2648, 2984, 3584, 3656, 3752, 3848, 3896, 3904, 3992, 4328, 4424, 4568, 4904, 5624, 5672, 5912, 6008, 6104, 6584, 6968, 7016, 7256, 7352, 7928, 8216, 8264, 8456, 8696, 8896, 8936, 9032, 9128, 9176, 9368, 9608, 9704, 10184, 10376, 10616, 10808, 11048, 11336, 11384, 11624, 12008, 12392, 12632, 12728, 12968, 13304, 13504, 13976, 14072, 14312, 14408, 14472, 14648, 14984, 15512, 15704, 15992, 16088, 16424, 16568, 16616, 16664, 16952, 17064, 17096, 17432, 17768, 18008, 18056, 18344, 18536, 18776, 19016, 19112, 19592, 19784, 19832, 20024, 20072, 20456, 20888, 21368, 21464, 21608, 21704, 22136, 22376, 22808, 22952, 23048, 23384, 23488, 23872, 24152, 24392, 24488, 24776, 25496, 25592, 25736, 25832, 26072, 26168, 26408, 26504, 26744

The OEIS comments state that "odd numbers with this property are much more numerous" and this is indeed true. In the same range (up to 26744) there are 2462 such odd numbers. Checking for \(n=26744\), we find that \( n^2=715241536 \) and we have: $$ \frac{\sigma(715241536,1)}{\sigma(715241536, 0)}=\frac{1419732111}{21}=67606291$$It can be noted that 26744 is itself an arithmetic number but this is hardly surprising as most numbers are and this includes all odd primes. To see this, take any odd prime \(p\) with the sum of its two divisors being \(1+p\), an even number and thus divisible by 2. According to Numbers Aplenty, \(p^p\) is also an arithmetic number.

It is possible to find quite long runs of consecutive arithmetic numbers. For example, one of length 105 starts at 3033935561. The smallest 3 × 3 magic square made of consecutive arithmetic numbers is shown below:

The 10000th arithmetic number is 12953. This gives a good idea of their frequency and why numbers that aren't arithmetic numbers are more interesting because they are less frequent. For example, \(84=2^2 \times 3 \times 7\) has twelve divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 that sum to 224. It is not an arithmetic number because 12 does not divide 224 without remainder.

Just to confuse matters, \(26744 = 2^3 \times 3343\) is a tau number, defined as a number that is divisible by the number of its divisors (in this case, eight). If we put all this together and ask what even numbers \(n\) have the properties that:$$ \begin{align} \frac{ \sigma(n^2,1)}{\sigma(n^2,0)}&=k_1\\ \frac{ \sigma(n,1)}{\sigma(n,0)}&=k_2\\ \frac{ n}{\sigma(n,0)}&=k_3 \end{align}$$where \(k_1, k_2, k_3\) are integers. It turns that the list is fairly short. In the range up to 40000, there are only 29 such numbers: 

1, 632, 1208, 3896, 6008, 6584, 7352, 7928, 8696, 10616, 11384, 13304, 14072, 14648, 15992, 20024, 21368, 22136, 26168, 26744, 28088, 28856, 30776, 31544, 32888, 35576, 36152, 38264, 39608

Barely Abundant Numbers

Today's diurnal age of 26742 days threw up an interesting number property that I can't recall coming across before. This is not surprising because the last occurrence occurred when I was 17816 days old, long before I started keeping track of the numbers associated with my diurnal age. The property in question qualifies both numbers for inclusion in OEIS A071927:

A071927

Barely abundant numbers: abundant \(n\) such that \( \dfrac{\sigma(n)}{n }< \dfrac{\sigma(m)}{m} \)for all abundant numbers \(m<n,\) \( \sigma(n) \) being the sum of the divisors of \(n\).

The terms in the sequence up to one million are:

12, 18, 20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 26742, 26778, 26886, 26898, 26958, 27042, 27078, 27102, 27114, 27138, 27282, 27294, 27366, 27402, 27498, 27546, 27582, 27618, 27726, 27822, 27834, 27858, 27894, 27906, 27942, 27978, 28038, 28074, 28146, 28218, 28326, 28338, 28374, 28398, 28506, 28554, 28698, 28722, 28734, 28758, 28794, 28806, 28878, 28902, 28986, 29166, 29226, 29262, 29334, 29418, 29454, 29514, 29586, 29598, 29622, 29658, 29706, 29742, 29802, 29814, 29838, 29922, 29958, 29994, 30018, 30054, 30066, 30126, 30138, 30234, 30306, 30354, 30462, 30486, 30522, 30594, 30606, 30642, 30678, 30714, 30882, 30918, 31002, 31026, 31074, 31134, 31182, 31254, 31362, 31386, 31398, 31422, 31566, 31638, 31674, 31686, 31782, 31818, 31854, 31938, 31998, 32082, 32106, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752

Figure 1 shows a plot of these values using a vertical log axis and the long evenly spaced stretch from 26742 to 32128 stands out clearly.

Figure 1: permalink

Figure 2 shows the same numbers showing the sigma(n)/n ratios and the factorisation (click on the image to enlarge):

Figure 2: permalink

It can be seen that all the barely abundant numbers from 26742 to 32128 are sphenic numbers of the form 2 x 3 x prime. Interesting all these numbers are admirable numbers, that is numbers whose proper factors add to the number when one of the factors is made negative. With these numbers the factor to be made negative is always 6. For example, the factors 1, 2, 3, -6, 4457, 8914, 13371  add to the admirable number 26742 and the factors 1, 2, 3, -6, 5039, 10078, 15117 add to the admirable number 30234.

It's easy to see why the 2 x 3 x prime are so successful in forming barely abundant numbers. The divisors of a number \(n\) in that case are \(2, 3, n/6, n/3, n/2, n\) and the sum of these divisors is \(2n+5\), giving a \( \sigma(n)/n \) ratio of \(2+5/n \). As \(n\) gets larger, the ratio gets smaller as can be seen in the progressive decrease in the size of the ratio from 26742 to 32128. Why the abrupt gap occurs after 32128 I'm not sure.

So barely abundant numbers will be cropping up quite regularly for the next six to seven years until another "drought" of such numbers occurs between 32129 and 77743.

Squarefree Semiprime Chains

The number associated with my diurnal age today, 26734, has an interesting property that qualifies it for membership in OEIS A177215:

A177215 Numbers \(k\) that are the products of two distinct primes such that \(2k-1, 4k-3, 8k-7, 16k-15, 32k-31 \text{ and } 64k-63\) are also products of two distinct primes.

Let's confirm that:

There are 281 members of OEIS  A177215 up to one million. I'll use the highest of them, 998185, as an example. It has the properties shown below:

What if we raise the stakes so to speak and apply the additional criterion that \(128k-123\) is also a semiprime with two distinct prime factors (in other words it's squarefree)? Well, the number shrinks to 81 in the range up to one million. The highest number that qualifies in that range is 991237 with the following properties:

Let's push thing further. How many numbers qualify once we add the additional criterion of \(256k-255\) being a squarefree semiprime? The number now reduces to 29, the highest of which remains our 991237. The revised table is shown below:

Let's keep pushing. How many numbers quality if we add the additional criteria that \(512k-511\) must be a squarefree semiprime? The number is now down to 14 and 991237 is still at the top. Here's the revised table.

Pushing further to include \(1024k-1023\), we find that only four numbers qualify in the range up to one million. These are 173311, 346621, 464245 and 563326. The table below shows the properties for 563326:

Proceeding logically, we now look at the criterion \(2056k-2055\) and we find that there is only one man left standing and that is 173311. Its table properties is shown below:

So overall, an interesting journey that ends with 173311 because this number doesn't pass the \(4096k-4095\) test. Of course, the journey never ends. We must ask what is the smallest number greater than one million that satisfies the criterion that \(4096k-4095\) is a squarefree semiprime? Well that number is 2212801. It's properties are shown below:

I'll stop there but here is a SageMathCell permalink to the program that I was using to discover these numbers. Feel free to experiment further.

ADDENDUM:

After making this post, I discovered that I'd covered much of the same territory in a post from October 17th 2018 titled Semiprime Chains. It will teach me to look back over my previous posts before posting. I've made so many posts of the years that it's easy to forget my earlier posts. One thing that is different between the two posts is the code that I used. The latter code is far more succinct. It's just as well that I looked back because the embedded code that I'd used was outdated. It contained the superseded print \(n\) command instead of the Python 3 print(\(n\)) command. I've corrected that now.

Here are the links to my previous posts on semiprimes:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• Semiprime Factor Ratios
• More about Golden Semiprimes
• 2019: A Numerical Profile
• Semiprime Chains
• Another Look at Semiprimes
• Visualisation of Semiprimes
• Semiprimes that Approximate Whole Numbers

I'm surprised at how many posts I've made over the years about semiprimes. It would be interesting to make a small booklet on the topic given the amount of material that I've accumulated.