Sunday, October 28

What's so special about 102735?

There are 8877691 nonnegative integers with distinct decimal digits. Last week, Eric Angelini suggested a procedure for numbers that do contain duplicated digits to be reduced to terms of this sequence simply by erasing all digits that appear more than once and bunching up the digits that remain. Leading zeros are of course ignored and we will posit that integers that disappear entirely become 0, so as to remain in the aforementioned A010784 set.

Iterating some starting integer by successive multiplication of 2, 3, 4, 5, 6, etc. and reducing at each turn, if necessary (as per the above), one hopes for a large number of turns before reaching 0. For example, starting with 24603:

2 * 24603 = 49206
3 * 49206 = [147618] => 4768
4 * 4768 = 19072
5 * 19072 = 95360
6 * 95360 = 572160
7 * 572160 = [4005120] => 4512
8 * 4512 = [36096] => 309
9 * 309 = 2781
10 * 2781 = 27810
11 * 27810 = [305910] => 3591
12 * 3591 = 43092
13 * 43092 = [560196] => 5019
14 * 5019 = [70266] => 702
15 * 702 = [10530] => 153
16 * 153 = [2448] => 28
17 * 28 = 476
18 * 476 = [8568] => 56
19 * 56 = 1064
20 * 1064 = [21280] => 180
21 * 180 = 3780
22 * 3780 = 83160
23 * 83160 = [1912680] => 92680
24 * 92680 = [2224320] => 430
25 * 430 = [10750] => 175
26 * 175 = [4550] => 40
27 * 40 = [1080] => 18
28 * 18 = 504
29 * 504 = [14616] => 4
30 * 4 = 120
31 * 120 = 3720
32 * 3720 = [119040] => 94
33 * 94 = 3102
34 * 3102 = 105468
35 * 105468 = [3691380] => 69180
36 * 69180 = [2490480] => 298
37 * 298 = [11026] => 26
38 * 26 = [988] => 9
39 * 9 = 351
40 * 351 = [14040] => 1
41 * 1 = 41
42 * 41 = [1722] => 17
43 * 17 = 731
44 * 731 = 32164
45 * 32164 = [1447380] => 17380
46 * 17380 = [799480] => 7480
47 * 7480 = [351560] => 3160
48 * 3160 = [151680] => 5680
49 * 5680 = [278320] => 7830
50 * 7830 = [391500] => 3915
51 * 3915 = [199665] => 15
52 * 15 = 780
53 * 780 = [41340] => 130
54 * 130 = [7020] => 72
55 * 72 = 3960
56 * 3960 = [221760] => 1760
57 * 1760 = [100320] => 132
58 * 132 = [7656] => 75
59 * 75 = [4425] => 25
60 * 25 = [1500] => 15
61 * 15 = 915
62 * 915 = 56730
63 * 56730 = [3573990] => 570
64 * 570 = 36480
65 * 36480 = [2371200] => 371
66 * 371 = [24486] => 286
67 * 286 = [19162] => 962
68 * 962 = [65416] => 541
69 * 541 = [37329] => 729
70 * 729 = [51030] => 513
71 * 513 = [36423] => 642
72 * 642 = [46224] => 6
73 * 6 = 438
74 * 438 = [32412] => 341
75 * 341 = [25575] => 27
76 * 27 = [2052] => 5
77 * 5 = 385
78 * 385 = [30030] => 0
79 * 0 = 0

Here, numbers that contain duplicated digits are shown in [square brackets] => followed by the reduced number. If we replace the iterated multiplication of successive integers with the iterated multiplication of successive primes, we can go much further. 102735 is the smallest integer with distinct decimal digits that generates the longest such chain:

2 * 102735 = [205470] => 2547
3 * 2547 = 7641
5 * 7641 = 38205
7 * 38205 = 267435
11 * 267435 = 2941785
13 * 2941785 = [38243205] => 8405
17 * 8405 = [142885] => 1425
19 * 1425 = [27075] => 205
23 * 205 = 4715
29 * 4715 = [136735] => 1675
31 * 1675 = [51925] => 192
37 * 192 = 7104
41 * 7104 = [291264] => 9164
43 * 9164 = 394052
47 * 394052 = [18520444] => 18520
53 * 18520 = 981560
59 * 981560 = [57912040] => 579124
61 * 579124 = [35326564] => 24
67 * 24 = 1608
71 * 1608 = [114168] => 468
73 * 468 = [34164] => 316
79 * 316 = [24964] => 296
83 * 296 = 24568
89 * 24568 = [2186552] => 186
97 * 186 = 18042
101 * 18042 = [1822242] => 184
103 * 184 = 18952
107 * 18952 = [2027864] => 7864
109 * 7864 = [857176] => 8516
113 * 8516 = 962308
127 * 962308 = [122213116] => 36
131 * 36 = 4716
137 * 4716 = [646092] => 4092
139 * 4092 = [568788] => 567
149 * 567 = [84483] => 3
151 * 3 = 453
157 * 453 = [71121] => 72
163 * 72 = [11736] => 736
167 * 736 = [122912] => 9
173 * 9 = [1557] => 17
179 * 17 = [3043] => 4
181 * 4 = 724
191 * 724 = [138284] => 1324
193 * 1324 = [255532] => 3
197 * 3 = 591
199 * 591 = [117609] => 7609
211 * 7609 = [1605499] => 16054
223 * 16054 = [3580042] => 35842
227 * 35842 = [8136134] => 864
229 * 864 = 197856
233 * 197856 = [46100448] => 618
239 * 618 = [147702] => 1402
241 * 1402 = [337882] => 72
251 * 72 = 18072
257 * 18072 = [4644504] => 650
263 * 650 = [170950] => 1795
269 * 1795 = [482855] => 42
271 * 42 = [11382] => 382
277 * 382 = [105814] => 584
281 * 584 = [164104] => 60
283 * 60 = 16980
293 * 16980 = [4975140] => 97510
307 * 97510 = [29935570] => 2370
311 * 2370 = [737070] => 3
313 * 3 = [939] => 3
317 * 3 = 951
331 * 951 = [314781] => 3478
337 * 3478 = [1172086] => 72086
347 * 72086 = [25013842] => 501384
349 * 501384 = [174983016] => 7498306
353 * 7498306 = [2646902018] => 4918
359 * 4918 = [1765562] => 172
367 * 172 = 63124
373 * 63124 = [23545252] => 34
379 * 34 = [12886] => 126
383 * 126 = [48258] => 425
389 * 425 = [165325] => 1632
397 * 1632 = [647904] => 6790
401 * 6790 = [2722790] => 90
409 * 90 = 36810
419 * 36810 = [15423390] => 154290
421 * 154290 = [64956090] => 45
431 * 45 = [19395] => 135
433 * 135 = [58455] => 84
439 * 84 = [36876] => 387
443 * 387 = [171441] => 7
449 * 7 = [3143] => 14
457 * 14 = 6398
461 * 6398 = [2949478] => 278
463 * 278 = [128714] => 2874
467 * 2874 = [1342158] => 34258
479 * 34258 = 16409582
487 * 16409582 = [7991466434] => 713
491 * 713 = [350083] => 58
499 * 58 = [28942] => 894
503 * 894 = [449682] => 9682
509 * 9682 = [4928138] => 49213
521 * 49213 = [25639973] => 2567
523 * 2567 = [1342541] => 325
541 * 325 = [175825] => 1782
547 * 1782 = [974754] => 95
557 * 95 = [52915] => 291
563 * 291 = [163833] => 168
569 * 168 = [95592] => 2
571 * 2 = [1142] => 42
577 * 42 = [24234] => 3
587 * 3 = [1761] => 76
593 * 76 = 45068
599 * 45068 = [26995732] => 6573
601 * 6573 = [3950373] => 9507
607 * 9507 = [5770749] => 5049
613 * 5049 = [3095037] => 957
617 * 957 = [590469] => 5046
619 * 5046 = [3123474] => 127
631 * 127 = 80137
641 * 80137 = [51367817] => 5368
643 * 5368 = [3451624] => 35162
647 * 35162 = [22749814] => 7981
653 * 7981 = [5211593] => 293
659 * 293 = 193087
661 * 193087 = [127630507] => 12635
673 * 12635 = [8503355] => 80
677 * 80 = 54160
683 * 54160 = [36991280] => 361280
691 * 361280 = [249644480] => 29680
701 * 29680 = [20805680] => 256
709 * 256 = [181504] => 8504
719 * 8504 = [6114376] => 437
727 * 437 = [317699] => 3176
733 * 3176 = [2328008] => 3
739 * 3 = [2217] => 17
743 * 17 = [12631] => 263
751 * 263 = [197513] => 9753
757 * 9753 = [7383021] => 78021
761 * 78021 = [59373981] => 5781
769 * 5781 = [4445589] => 89
773 * 89 = [68797] => 689
787 * 689 = [542243] => 53
797 * 53 = [42241] => 1
809 * 1 = 809
811 * 809 = [656099] => 50
821 * 50 = [41050] => 415
823 * 415 = [341545] => 31
827 * 31 = 25637
829 * 25637 = [21253073] => 1507
839 * 1507 = [1264373] => 12647
853 * 12647 = [10787891] => 9
857 * 9 = [7713] => 13
859 * 13 = [11167] => 67
863 * 67 = 57821
877 * 57821 = [50709017] => 591
881 * 591 = 520671
883 * 520671 = [459752493] => 723
887 * 723 = [641301] => 6430
907 * 6430 = [5832010] => 58321
911 * 58321 = [53130431] => 504
919 * 504 = [463176] => 4317
929 * 4317 = [4010493] => 193
937 * 193 = [180841] => 4
941 * 4 = 3764
947 * 3764 = [3564508] => 36408
953 * 36408 = [34696824] => 3982
967 * 3982 = [3850594] => 38094
971 * 38094 = [36989274] => 368274
977 * 368274 = [359803698] => 506
983 * 506 = [497398] => 4738
991 * 4738 = [4695358] => 46938
997 * 46938 = [46797186] => 4918
1009 * 4918 = [4962262] => 49
1013 * 49 = 49637
1019 * 49637 = [50580103] => 813
1021 * 813 = [830073] => 87
1031 * 87 = [89697] => 867
1033 * 867 = [895611] => 8956
1039 * 8956 = 9305284
1049 * 9305284 = [9761242916] => 74
1051 * 74 = [77774] => 4
1061 * 4 = [4244] => 2
1063 * 2 = [2126] => 16
1069 * 16 = [17104] => 704
1087 * 704 = 765248
1091 * 765248 = [834885568] => 346
1093 * 346 = [378178] => 31
1097 * 31 = [34007] => 347
1103 * 347 = 382741
1109 * 382741 = [424459769] => 2576
1117 * 2576 = [2877392] => 839
1123 * 839 = [942197] => 4217
1129 * 4217 = [4760993] => 47603
1151 * 47603 = [54791053] => 479103
1153 * 479103 = [552405759] => 24079
1163 * 24079 = [28003877] => 23
1171 * 23 = [26933] => 269
1181 * 269 = 317689
1187 * 317689 = [377096843] => 9684
1193 * 9684 = [11553012] => 302
1201 * 302 = [362702] => 3670
1213 * 3670 = [4451710] => 570
1217 * 570 = [693690] => 30
1223 * 30 = [36690] => 390
1229 * 390 = 479310
1231 * 479310 = [590030610] => 59361
1237 * 59361 = [73429557] => 3429
1249 * 3429 = [4282821] => 41
1259 * 41 = [51619] => 569
1277 * 569 = [726613] => 7213
1279 * 7213 = [9225427] => 9547
1283 * 9547 = [12248801] => 40
1289 * 40 = [51560] => 160
1291 * 160 = [206560] => 25
1297 * 25 = [32425] => 345
1301 * 345 = [448845] => 5
1303 * 5 = [6515] => 61
1307 * 61 = [79727] => 92
1319 * 92 = [121348] => 2348
1321 * 2348 = [3101708] => 378
1327 * 378 = [501606] => 51
1361 * 51 = [69411] => 694
1367 * 694 = [948698] => 46
1373 * 46 = 63158
1381 * 63158 = [87221198] => 79
1399 * 79 = [110521] => 52
1409 * 52 = 73268
1423 * 73268 = [104260364] => 123
1427 * 123 = [175521] => 72
1429 * 72 = [102888] => 102
1433 * 102 = [146166] => 4
1439 * 4 = [5756] => 76
1447 * 76 = [109972] => 1072
1451 * 1072 = [1555472] => 1472
1453 * 1472 = [2138816] => 236
1459 * 236 = [344324] => 2
1471 * 2 = [2942] => 94
1481 * 94 = [139214] => 3924
1483 * 3924 = [5819292] => 581
1487 * 581 = 863947
1489 * 863947 = [1286417083] => 264703
1493 * 264703 = [395201579] => 32017
1499 * 32017 = [47993483] => 78
1511 * 78 = [117858] => 75
1523 * 75 = [114225] => 45
1531 * 45 = [68895] => 695
1543 * 695 = 1072385
1549 * 1072385 = [1661124365] => 2435
1553 * 2435 = [3781555] => 3781
1559 * 3781 = [5894579] => 847
1567 * 847 = [1327249] => 13749
1571 * 13749 = [21599679] => 21567
1579 * 21567 = [34054293] => 529
1583 * 529 = [837407] => 8340
1597 * 8340 = [13318980] => 90
1601 * 90 = [144090] => 19
1607 * 19 = [30533] => 5
1609 * 5 = 8045
1613 * 8045 = [12976585] => 129768
1619 * 129768 = [210094392] => 143
1621 * 143 = [231803] => 2180
1627 * 2180 = [3546860] => 35480
1637 * 35480 = [58080760] => 576
1657 * 576 = [954432] => 9532
1663 * 9532 = [15851716] => 876
1667 * 876 = [1460292] => 14609
1669 * 14609 = [24382421] => 381
1693 * 381 = [645033] => 6450
1697 * 6450 = [10945650] => 1946
1699 * 1946 = [3306254] => 6254
1709 * 6254 = [10688086] => 1
1721 * 1 = [1721] => 72
1723 * 72 = 124056
1733 * 124056 = [214989048] => 210
1741 * 210 = [365610] => 3510
1747 * 3510 = [6131970] => 63970
1753 * 63970 = [112139410] => 23940
1759 * 23940 = [42110460] => 26
1777 * 26 = [46202] => 460
1783 * 460 = [820180] => 21
1787 * 21 = [37527] => 352
1789 * 352 = [629728] => 6978
1801 * 6978 = [12567378] => 125638
1811 * 125638 = [227530418] => 7530418
1823 * 7530418 = [13727952014] => 39504
1831 * 39504 = [72331824] => 7184
1847 * 7184 = [13268848] => 13264
1861 * 13264 = [24684304] => 26830
1867 * 26830 = [50091610] => 596
1871 * 596 = [1115116] => 56
1873 * 56 = [104888] => 104
1877 * 104 = 195208
1879 * 195208 = [366795832] => 79582
1889 * 79582 = [150330398] => 1598
1901 * 1598 = [3037798] => 98
1907 * 98 = [186886] => 1
1913 * 1 = [1913] => 93
1931 * 93 = 179583
1933 * 179583 = [347133939] => 471
1949 * 471 = [917979] => 1
1951 * 1 = [1951] => 95
1973 * 95 = 187435
1979 * 187435 = [370933865] => 709865
1987 * 709865 = [1410501755] => 47
1993 * 47 = 93671
1997 * 93671 = [187060987] => 169
1999 * 169 = [337831] => 781
2003 * 781 = [1564343] => 156
2011 * 156 = [313716] => 76
2017 * 76 = [153292] => 1539
2027 * 1539 = [3119553] => 9
2029 * 9 = [18261] => 826
2039 * 826 = [1684214] => 682
2053 * 682 = [1400146] => 6
2063 * 6 = 12378
2069 * 12378 = [25610082] => 5618
2081 * 5618 = [11691058] => 69058
2083 * 69058 = [143847814] => 37
2087 * 37 = [77219] => 219
2089 * 219 = [457491] => 5791
2099 * 5791 = [12155309] => 2309
2111 * 2309 = [4874299] => 872
2113 * 872 = 1842536
2129 * 1842536 = [3922759144] => 3751
2131 * 3751 = [7993381] => 781
2137 * 781 = [1668997] => 187
2141 * 187 = [400367] => 4367
2143 * 4367 = [9358481] => 93541
2153 * 93541 = [201393773] => 2019
2161 * 2019 = [4363059] => 46059
2179 * 46059 = [100362561] => 325
2203 * 325 = [715975] => 19
2207 * 19 = [41933] => 419
2213 * 419 = [927247] => 94
2221 * 94 = [208774] => 2084
2237 * 2084 = [4661908] => 41908
2239 * 41908 = [93832012] => 9801
2243 * 9801 = [21983643] => 219864
2251 * 219864 = [494913864] => 1386
2267 * 1386 = [3142062] => 31406
2269 * 31406 = [71260214] => 7604
2273 * 7604 = [17283892] => 1739
2281 * 1739 = [3966659] => 35
2287 * 35 = [80045] => 845
2293 * 845 = [1937585] => 19378
2297 * 19378 = [44511266] => 52
2309 * 52 = [120068] => 1268
2311 * 1268 = [2930348] => 29048
2333 * 29048 = [67768984] => 94
2339 * 94 = [219866] => 2198
2341 * 2198 = [5145518] => 48
2347 * 48 = [112656] => 25
2351 * 25 = [58775] => 8
2357 * 8 = [18856] => 156
2371 * 156 = [369876] => 3987
2377 * 3987 = [9477099] => 40
2381 * 40 = 95240
2383 * 95240 = [226956920] => 50
2389 * 50 = [119450] => 9450
2393 * 9450 = [22613850] => 613850
2399 * 613850 = [1472626150] => 4750
2411 * 4750 = [11452250] => 40
2417 * 40 = [96680] => 980
2423 * 980 = [2374540] => 23750
2437 * 23750 = [57878750] => 0
2441 * 0 = 0

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