## Sunday, October 28, 2018

### 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 =  => 4768
4 * 4768 = 19072
5 * 19072 = 95360
6 * 95360 = 572160
7 * 572160 =  => 4512
8 * 4512 =  => 309
9 * 309 = 2781
10 * 2781 = 27810
11 * 27810 =  => 3591
12 * 3591 = 43092
13 * 43092 =  => 5019
14 * 5019 =  => 702
15 * 702 =  => 153
16 * 153 =  => 28
17 * 28 = 476
18 * 476 =  => 56
19 * 56 = 1064
20 * 1064 =  => 180
21 * 180 = 3780
22 * 3780 = 83160
23 * 83160 =  => 92680
24 * 92680 =  => 430
25 * 430 =  => 175
26 * 175 =  => 40
27 * 40 =  => 18
28 * 18 = 504
29 * 504 =  => 4
30 * 4 = 120
31 * 120 = 3720
32 * 3720 =  => 94
33 * 94 = 3102
34 * 3102 = 105468
35 * 105468 =  => 69180
36 * 69180 =  => 298
37 * 298 =  => 26
38 * 26 =  => 9
39 * 9 = 351
40 * 351 =  => 1
41 * 1 = 41
42 * 41 =  => 17
43 * 17 = 731
44 * 731 = 32164
45 * 32164 =  => 17380
46 * 17380 =  => 7480
47 * 7480 =  => 3160
48 * 3160 =  => 5680
49 * 5680 =  => 7830
50 * 7830 =  => 3915
51 * 3915 =  => 15
52 * 15 = 780
53 * 780 =  => 130
54 * 130 =  => 72
55 * 72 = 3960
56 * 3960 =  => 1760
57 * 1760 =  => 132
58 * 132 =  => 75
59 * 75 =  => 25
60 * 25 =  => 15
61 * 15 = 915
62 * 915 = 56730
63 * 56730 =  => 570
64 * 570 = 36480
65 * 36480 =  => 371
66 * 371 =  => 286
67 * 286 =  => 962
68 * 962 =  => 541
69 * 541 =  => 729
70 * 729 =  => 513
71 * 513 =  => 642
72 * 642 =  => 6
73 * 6 = 438
74 * 438 =  => 341
75 * 341 =  => 27
76 * 27 =  => 5
77 * 5 = 385
78 * 385 =  => 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 =  => 2547
3 * 2547 = 7641
5 * 7641 = 38205
7 * 38205 = 267435
11 * 267435 = 2941785
13 * 2941785 =  => 8405
17 * 8405 =  => 1425
19 * 1425 =  => 205
23 * 205 = 4715
29 * 4715 =  => 1675
31 * 1675 =  => 192
37 * 192 = 7104
41 * 7104 =  => 9164
43 * 9164 = 394052
47 * 394052 =  => 18520
53 * 18520 = 981560
59 * 981560 =  => 579124
61 * 579124 =  => 24
67 * 24 = 1608
71 * 1608 =  => 468
73 * 468 =  => 316
79 * 316 =  => 296
83 * 296 = 24568
89 * 24568 =  => 186
97 * 186 = 18042
101 * 18042 =  => 184
103 * 184 = 18952
107 * 18952 =  => 7864
109 * 7864 =  => 8516
113 * 8516 = 962308
127 * 962308 =  => 36
131 * 36 = 4716
137 * 4716 =  => 4092
139 * 4092 =  => 567
149 * 567 =  => 3
151 * 3 = 453
157 * 453 =  => 72
163 * 72 =  => 736
167 * 736 =  => 9
173 * 9 =  => 17
179 * 17 =  => 4
181 * 4 = 724
191 * 724 =  => 1324
193 * 1324 =  => 3
197 * 3 = 591
199 * 591 =  => 7609
211 * 7609 =  => 16054
223 * 16054 =  => 35842
227 * 35842 =  => 864
229 * 864 = 197856
233 * 197856 =  => 618
239 * 618 =  => 1402
241 * 1402 =  => 72
251 * 72 = 18072
257 * 18072 =  => 650
263 * 650 =  => 1795
269 * 1795 =  => 42
271 * 42 =  => 382
277 * 382 =  => 584
281 * 584 =  => 60
283 * 60 = 16980
293 * 16980 =  => 97510
307 * 97510 =  => 2370
311 * 2370 =  => 3
313 * 3 =  => 3
317 * 3 = 951
331 * 951 =  => 3478
337 * 3478 =  => 72086
347 * 72086 =  => 501384
349 * 501384 =  => 7498306
353 * 7498306 =  => 4918
359 * 4918 =  => 172
367 * 172 = 63124
373 * 63124 =  => 34
379 * 34 =  => 126
383 * 126 =  => 425
389 * 425 =  => 1632
397 * 1632 =  => 6790
401 * 6790 =  => 90
409 * 90 = 36810
419 * 36810 =  => 154290
421 * 154290 =  => 45
431 * 45 =  => 135
433 * 135 =  => 84
439 * 84 =  => 387
443 * 387 =  => 7
449 * 7 =  => 14
457 * 14 = 6398
461 * 6398 =  => 278
463 * 278 =  => 2874
467 * 2874 =  => 34258
479 * 34258 = 16409582
487 * 16409582 =  => 713
491 * 713 =  => 58
499 * 58 =  => 894
503 * 894 =  => 9682
509 * 9682 =  => 49213
521 * 49213 =  => 2567
523 * 2567 =  => 325
541 * 325 =  => 1782
547 * 1782 =  => 95
557 * 95 =  => 291
563 * 291 =  => 168
569 * 168 =  => 2
571 * 2 =  => 42
577 * 42 =  => 3
587 * 3 =  => 76
593 * 76 = 45068
599 * 45068 =  => 6573
601 * 6573 =  => 9507
607 * 9507 =  => 5049
613 * 5049 =  => 957
617 * 957 =  => 5046
619 * 5046 =  => 127
631 * 127 = 80137
641 * 80137 =  => 5368
643 * 5368 =  => 35162
647 * 35162 =  => 7981
653 * 7981 =  => 293
659 * 293 = 193087
661 * 193087 =  => 12635
673 * 12635 =  => 80
677 * 80 = 54160
683 * 54160 =  => 361280
691 * 361280 =  => 29680
701 * 29680 =  => 256
709 * 256 =  => 8504
719 * 8504 =  => 437
727 * 437 =  => 3176
733 * 3176 =  => 3
739 * 3 =  => 17
743 * 17 =  => 263
751 * 263 =  => 9753
757 * 9753 =  => 78021
761 * 78021 =  => 5781
769 * 5781 =  => 89
773 * 89 =  => 689
787 * 689 =  => 53
797 * 53 =  => 1
809 * 1 = 809
811 * 809 =  => 50
821 * 50 =  => 415
823 * 415 =  => 31
827 * 31 = 25637
829 * 25637 =  => 1507
839 * 1507 =  => 12647
853 * 12647 =  => 9
857 * 9 =  => 13
859 * 13 =  => 67
863 * 67 = 57821
877 * 57821 =  => 591
881 * 591 = 520671
883 * 520671 =  => 723
887 * 723 =  => 6430
907 * 6430 =  => 58321
911 * 58321 =  => 504
919 * 504 =  => 4317
929 * 4317 =  => 193
937 * 193 =  => 4
941 * 4 = 3764
947 * 3764 =  => 36408
953 * 36408 =  => 3982
967 * 3982 =  => 38094
971 * 38094 =  => 368274
977 * 368274 =  => 506
983 * 506 =  => 4738
991 * 4738 =  => 46938
997 * 46938 =  => 4918
1009 * 4918 =  => 49
1013 * 49 = 49637
1019 * 49637 =  => 813
1021 * 813 =  => 87
1031 * 87 =  => 867
1033 * 867 =  => 8956
1039 * 8956 = 9305284
1049 * 9305284 =  => 74
1051 * 74 =  => 4
1061 * 4 =  => 2
1063 * 2 =  => 16
1069 * 16 =  => 704
1087 * 704 = 765248
1091 * 765248 =  => 346
1093 * 346 =  => 31
1097 * 31 =  => 347
1103 * 347 = 382741
1109 * 382741 =  => 2576
1117 * 2576 =  => 839
1123 * 839 =  => 4217
1129 * 4217 =  => 47603
1151 * 47603 =  => 479103
1153 * 479103 =  => 24079
1163 * 24079 =  => 23
1171 * 23 =  => 269
1181 * 269 = 317689
1187 * 317689 =  => 9684
1193 * 9684 =  => 302
1201 * 302 =  => 3670
1213 * 3670 =  => 570
1217 * 570 =  => 30
1223 * 30 =  => 390
1229 * 390 = 479310
1231 * 479310 =  => 59361
1237 * 59361 =  => 3429
1249 * 3429 =  => 41
1259 * 41 =  => 569
1277 * 569 =  => 7213
1279 * 7213 =  => 9547
1283 * 9547 =  => 40
1289 * 40 =  => 160
1291 * 160 =  => 25
1297 * 25 =  => 345
1301 * 345 =  => 5
1303 * 5 =  => 61
1307 * 61 =  => 92
1319 * 92 =  => 2348
1321 * 2348 =  => 378
1327 * 378 =  => 51
1361 * 51 =  => 694
1367 * 694 =  => 46
1373 * 46 = 63158
1381 * 63158 =  => 79
1399 * 79 =  => 52
1409 * 52 = 73268
1423 * 73268 =  => 123
1427 * 123 =  => 72
1429 * 72 =  => 102
1433 * 102 =  => 4
1439 * 4 =  => 76
1447 * 76 =  => 1072
1451 * 1072 =  => 1472
1453 * 1472 =  => 236
1459 * 236 =  => 2
1471 * 2 =  => 94
1481 * 94 =  => 3924
1483 * 3924 =  => 581
1487 * 581 = 863947
1489 * 863947 =  => 264703
1493 * 264703 =  => 32017
1499 * 32017 =  => 78
1511 * 78 =  => 75
1523 * 75 =  => 45
1531 * 45 =  => 695
1543 * 695 = 1072385
1549 * 1072385 =  => 2435
1553 * 2435 =  => 3781
1559 * 3781 =  => 847
1567 * 847 =  => 13749
1571 * 13749 =  => 21567
1579 * 21567 =  => 529
1583 * 529 =  => 8340
1597 * 8340 =  => 90
1601 * 90 =  => 19
1607 * 19 =  => 5
1609 * 5 = 8045
1613 * 8045 =  => 129768
1619 * 129768 =  => 143
1621 * 143 =  => 2180
1627 * 2180 =  => 35480
1637 * 35480 =  => 576
1657 * 576 =  => 9532
1663 * 9532 =  => 876
1667 * 876 =  => 14609
1669 * 14609 =  => 381
1693 * 381 =  => 6450
1697 * 6450 =  => 1946
1699 * 1946 =  => 6254
1709 * 6254 =  => 1
1721 * 1 =  => 72
1723 * 72 = 124056
1733 * 124056 =  => 210
1741 * 210 =  => 3510
1747 * 3510 =  => 63970
1753 * 63970 =  => 23940
1759 * 23940 =  => 26
1777 * 26 =  => 460
1783 * 460 =  => 21
1787 * 21 =  => 352
1789 * 352 =  => 6978
1801 * 6978 =  => 125638
1811 * 125638 =  => 7530418
1823 * 7530418 =  => 39504
1831 * 39504 =  => 7184
1847 * 7184 =  => 13264
1861 * 13264 =  => 26830
1867 * 26830 =  => 596
1871 * 596 =  => 56
1873 * 56 =  => 104
1877 * 104 = 195208
1879 * 195208 =  => 79582
1889 * 79582 =  => 1598
1901 * 1598 =  => 98
1907 * 98 =  => 1
1913 * 1 =  => 93
1931 * 93 = 179583
1933 * 179583 =  => 471
1949 * 471 =  => 1
1951 * 1 =  => 95
1973 * 95 = 187435
1979 * 187435 =  => 709865
1987 * 709865 =  => 47
1993 * 47 = 93671
1997 * 93671 =  => 169
1999 * 169 =  => 781
2003 * 781 =  => 156
2011 * 156 =  => 76
2017 * 76 =  => 1539
2027 * 1539 =  => 9
2029 * 9 =  => 826
2039 * 826 =  => 682
2053 * 682 =  => 6
2063 * 6 = 12378
2069 * 12378 =  => 5618
2081 * 5618 =  => 69058
2083 * 69058 =  => 37
2087 * 37 =  => 219
2089 * 219 =  => 5791
2099 * 5791 =  => 2309
2111 * 2309 =  => 872
2113 * 872 = 1842536
2129 * 1842536 =  => 3751
2131 * 3751 =  => 781
2137 * 781 =  => 187
2141 * 187 =  => 4367
2143 * 4367 =  => 93541
2153 * 93541 =  => 2019
2161 * 2019 =  => 46059
2179 * 46059 =  => 325
2203 * 325 =  => 19
2207 * 19 =  => 419
2213 * 419 =  => 94
2221 * 94 =  => 2084
2237 * 2084 =  => 41908
2239 * 41908 =  => 9801
2243 * 9801 =  => 219864
2251 * 219864 =  => 1386
2267 * 1386 =  => 31406
2269 * 31406 =  => 7604
2273 * 7604 =  => 1739
2281 * 1739 =  => 35
2287 * 35 =  => 845
2293 * 845 =  => 19378
2297 * 19378 =  => 52
2309 * 52 =  => 1268
2311 * 1268 =  => 29048
2333 * 29048 =  => 94
2339 * 94 =  => 2198
2341 * 2198 =  => 48
2347 * 48 =  => 25
2351 * 25 =  => 8
2357 * 8 =  => 156
2371 * 156 =  => 3987
2377 * 3987 =  => 40
2381 * 40 = 95240
2383 * 95240 =  => 50
2389 * 50 =  => 9450
2393 * 9450 =  => 613850
2399 * 613850 =  => 4750
2411 * 4750 =  => 40
2417 * 40 =  => 980
2423 * 980 =  => 23750
2437 * 23750 =  => 0
2441 * 0 = 0

## Saturday, October 20, 2018

### M from heaven

Some twenty months ago I posted a snippet of a probable-prime leaderboard and my #44 position in it. I was looking forward back then to reaching position #40. A month-and-a-half ago I had reached position #35, at which I was still yesterday when I grabbed this:

On October 8, Joerg Arndt received a personal message from an "M" that gave five large probable primes with the intent of removing from a specific OEIS sequence five of its terms. Joerg shared the post with the Sequence Fanatics Discussion list the next day. Now it's very unusual to have an OEIS sequence that removes terms as knowledge is advanced and by October 10, Neil Sloane (owner of the OEIS) killed that sequence because it was likely intended to be a duplicate of A076337 — but with additional (unproven) terms. What M did was show that five of those extra terms didn't in fact belong.

Also on October 10, I emailed M to let him/her know that the five probable primes were not yet in the probable prime records database. On Thursday (October 18), M (finally) replied that he/she had no desire to receive credit and that if I wanted to submit the results I should "feel free". So I rechecked the five numbers for probable primality and — this morning — submitted them. A couple of hours ago the database was updated:

That was quite the ascent! I should point out that Norbert Schneider at #28 is the only other person who — like me — has been actively looking for new Leyland primes. Of the thirteen currently largest-known examples, Norbert has contributed eight.

## Monday, October 15, 2018

### The nickel

In late August, a bag of potatoes kept in the basement showed obvious signs of having been attacked by a rodent. It had been many, many years since we last dealt with mice in the house and the idea that there was an unknown entry point somewhere was not a notion that I wanted to entertain. Perhaps it was a one-off, finding its way through the drainage system. At any rate, I resolved to live-trap the creature and release it outside.

My first attempt at a trap was ill-conceived. The container was too light. The twelve-sided Fiji 50-cents coin providing access to the container, too stable. And the paper underlay, too tearable. By early September I had corrected my deficiencies. A heavier plant pot propped up by a round Canadian nickel on a small glass plate:

The bait was peanut butter smeared on the inner wall. The next morning I found the nickel beside the fallen pot and, turning the whole thing over, a mouse inside. After releasing it I reset the trap and, looking every few days, found no more disturbances that month.

Last Wednesday, Catherine went into the basement to check on an unrelated matter and accidentally sprung the trap. A closer inspection showed the peanut butter gone and mouse droppings on the table. Argh! So I set it up again and the next morning snapped this photo of my second capture:

Yesterday morning I had a third one! So when in the pre-dawn hours today I heard a sound in the basement, I was fairly apprehensive. The pot had dropped but there was no mouse inside. The nickel was not on the table and it was not anywhere obvious on the floor underneath but I was in no mood to search for it. A few hours later I checked my wallet for another nickel but I did not have one. I must ask Catherine for one when she comes down for lunch later!

It was a rainy morning and I left Bodie's walk a little later than normal. Near my usual juncture on the walk I prepared to cross the street.

As I stepped off the curb I noticed something shiny on the road. I wondered: What are the odds?

## Sunday, October 07, 2018

### Consecutive primes summing to a conspicuous prime

After Friday's "What else?" I decided to tackle Q2 on Carlos Rivera's PrimePuzzles webpage.

Here are three large primes, each the sum of three consecutive primes:

Leyland(54,7) = 54^7+7^54 <46 digits>

1439371522465478854678431032569243142159152897
1439371522465478854678431032569243142159152957
1439371522465478854678431032569243142159152979
4318114567396436564035293097707729426477458833

Cullen(141) = 141*2^141+1 <45 digits>

131016878041367410956522344851809123847634759
131016878041367410956522344851809123847634887
131016878041367410956522344851809123847635187
393050634124102232869567034555427371542904833

Mersenne(127) = 2^127-1 <39 digits>

56713727820156410577229101238628035201
56713727820156410577229101238628035243
56713727820156410577229101238628035283
170141183460469231731687303715884105727

And here's a smaller prime, the sum of 175 consecutive primes:

Mersenne(61) = 2^61-1 <19 digits>

13176245766932173
13176245766932207
13176245766932219
13176245766932231
13176245766932279
13176245766932321
13176245766932363
13176245766932411
13176245766932441
13176245766932473
13176245766932477
13176245766932497
13176245766932587
13176245766932639
13176245766932803
13176245766932819
13176245766932837
13176245766932867
13176245766932873
13176245766932887
13176245766932917
13176245766932951
13176245766932977
13176245766933017
13176245766933049
13176245766933079
13176245766933113
13176245766933121
13176245766933133
13176245766933269
13176245766933343
13176245766933407
13176245766933437
13176245766933481
13176245766933521
13176245766933523
13176245766933551
13176245766933583
13176245766933593
13176245766933749
13176245766933797
13176245766933803
13176245766933829
13176245766933883
13176245766933979
13176245766933983
13176245766934007
13176245766934027
13176245766934069
13176245766934111
13176245766934153
13176245766934177
13176245766934219
13176245766934297
13176245766934307
13176245766934361
13176245766934373
13176245766934379
13176245766934391
13176245766934457
13176245766934477
13176245766934489
13176245766934511
13176245766934543
13176245766934561
13176245766934661
13176245766934721
13176245766934769
13176245766934781
13176245766934787
13176245766934823
13176245766934841
13176245766934883
13176245766934931
13176245766934951
13176245766934961
13176245766934979
13176245766934981
13176245766935021
13176245766935039
13176245766935081
13176245766935123
13176245766935183
13176245766935323
13176245766935327
13176245766935347
13176245766935351
13176245766935357
13176245766935443
13176245766935477
13176245766935479
13176245766935501
13176245766935527
13176245766935569
13176245766935639
13176245766935663
13176245766935687
13176245766935731
13176245766935773
13176245766935819
13176245766935827
13176245766935891
13176245766935893
13176245766935941
13176245766935963
13176245766936019
13176245766936053
13176245766936059
13176245766936061
13176245766936163
13176245766936209
13176245766936229
13176245766936251
13176245766936281
13176245766936319
13176245766936331
13176245766936403
13176245766936407
13176245766936463
13176245766936521
13176245766936619
13176245766936643
13176245766936667
13176245766936691
13176245766936701
13176245766936709
13176245766936773
13176245766936821
13176245766936883
13176245766936953
13176245766936971
13176245766937013
13176245766937027
13176245766937069
13176245766937093
13176245766937111
13176245766937189
13176245766937217
13176245766937247
13176245766937301
13176245766937441
13176245766937553
13176245766937573
13176245766937607
13176245766937613
13176245766937691
13176245766937697
13176245766937739
13176245766937769
13176245766937777
13176245766937789
13176245766937793
13176245766937801
13176245766937831
13176245766937859
13176245766937871
13176245766937907
13176245766937927
13176245766937949
13176245766937951
13176245766937973
13176245766937987
13176245766938039
13176245766938071
13176245766938123
13176245766938131
13176245766938147
13176245766938161
13176245766938173
13176245766938183
13176245766938273
13176245766938291
13176245766938381
13176245766938447
13176245766938521
2305843009213693951

I also had a look at A067377 and decided to update it a bit. I have an indexed 293768 terms of A067377 going to 10^7, listing for each prime the possible number of consecutive primes into which it may be decomposed. A 41 MB .txt file (2565345 terms, indexing not included) takes us to 10^8. Here is a 13 MB .zip compression of that.

## Friday, October 05, 2018

### What else?

Last month's seven consecutive primes summing to a repunit prime has made it to a PrimePuzzles website, where my "what else?" is transmuted into Q1. I'll note here my own negative results.

The seven-consecutive-primes is the only solution that I found for R19 (the repunit consisting of 19 ones) up to 10000 consecutive primes. I found no solution for R23 up to 10000 consecutive primes. I found no solution for R317 up to 1000 consecutive primes. I found no solution for R1031 up to 500 consecutive primes. Finally, I can show that R49081 is not the sum of three consecutive primes:

Let k equal (R49081 minus 1) divided by 3. This integer is not prime. The largest prime less than k is ((10^49081-1)/9-1)/3-71387 and the smallest prime greater than k is ((10^49081-1)/9-1)/3+84453. These two primes are therefore consecutive and both are required to be part of the three consecutive primes that might sum to R49081. It's an easy matter now to determine that the third number needs to be ((10^49081-1)/9-1)/3-13065 which (alas) is between our two consecutive primes and is therefore composite.

By the way, if you think that the 155839 composites between the two (above) consecutive primes constitutes a big prime gap, you'd be wrong. The gap has a "merit" of only 1.379. A website that charts large prime gaps won't even consider gaps with a merit less than 10.