Why does this branching structure appear in consecutive 7 digit "double" palindromes?
Dots at the end of the rows indicate that I excluded following values for that row
(because of the character limit for the post - see the link below for full pattern)
(Scroll around)
3 :
4 :1325
5 :12
6 :8110 432 4
7 :100 4 164051
8 :122 564 414 6129
9 :1111421 136 644 81 5 167
10 :163 24 5 815 13 87 6128 118 5
11 : 187 6 19 11296 105 22106 720275 121
12 :218 6 21 145 6 126 16 8 126 3486 97 127 6 15 127
13 :249 742129 21 6 151 16 8 149 17 10555 79 7 148 818 150
14 :282 7198 22 7 1433418 7 176 18 187 18 442 17 7 174 18 8 176 17 188
15 :317 20214 352116936 9204 20 214 20 521 7 20519 8204 18 214 19 208 7
16 :355 20 243 60 194 8 23 24522 24422 268 2789 295 923621 243 21 2368 21 236 8
17 :395 21 274 20 267 9 25 236 64 27724 27724 3119 288 41 20 279 2929 276 12 10 266 91113 266 9 64
18 :437 21 308 22 300 9 2131 24624 43 337 31226 685 9 21 4 338 21 3059 326 9 30122 11 299 1011 14
19 :481 23 344 22 335 10 23 31224 9 29 23353 36511 36214 383 11 22 336 9 30 24 33910 24 339 9 362 10 336 23 12 334 22 15
20 :527 23 382 24 372 10 23 373 10 247 25 312 76 405 11 402 13 424 13 399 10 2436 333 3810 27 375 10 26 375 10 37624 11 373 24 13 372 23
21 :575 25 423 24 412 11 25 412 10 24 37 348 27 51 447 10 444 13 926 12 13 11 414 11 36 13 14 34667 10 1316 413 11 29413 10 415 25 12 412 25 456
22 :625 1312 465 26 454 11 25 454 11 26 428 27 11 36 30461 11 488 12 486 15 509 14 14 11 455 12 14 468 11 26 1213 15 454 121418 454 12484 12456 27 466 27
23 :677 1413 510 1412 498 12 27 498 11 26 47129 12 38 32 426 75 11 534 12 532 14 556 16528 14 14 512 11 16 56 452 39 27 17 498 26 516 51429 511 29 510 29 572
24 :731 14 12 557 1513 544 12 1413 544 12 28 51629 12 2846 468 30 44 12 33549 12 580 14 1203 16 14 557 13 16 557 12 18 38 16 544 27 561 28 563 559 31 557 31 591 621 28
25 :787 15 13 606 15 12 592 13 15 13 592 12 28 593 13 29563 30 12 29 13 35 596 13630 13 627 16656 32 606 15 15 604 13 18 605 31 7 28 608 29 609 28 57933 607 33 606 33 1321 29 5
26 :845 15 13 657 16 13 642 14 15 13 642 13 30 642 13 30612 32 13 451522 558 83 14682 14679 15710 695 32 655 1517 654 32 655 40 31 658 30 660 31 62835 657 35 657 35 726 695 29 7 2213
27 :905 16 14 710 16 13 694 15 16 14 694 13 1615 694 14 32663 32 13 85 606 33 48 54696 15733 1534 32 708 33 706 32 705 33 42 708 32 711 33 713 32 67937 710 37 749 782 30 694 17 16...
28 :967 16 14 765 17 14 748 16 16 14 748 14 17 15 748 14 3371634 14 748 33 14 32 3225 749 804 802 837 33 764 33 761 33 759 33 760 33 4 34 763 35 766 34 768 3573239 765 39 ...
29 :103117 15 822 17 14 804 17 17 15 804 14 17 15 804 15 171877134 14 804 35 15 67 25 708 91 59804 860 897 863 34 819 33 816 34 815 33 816 43 37 820 36 823 37 825 36...
30 :109717 15 881 18 15 862 18 17 15 862 15 18 15 862 15 17 18863 15 862 35 15 97 763 36 51 33 27862 921 1887 34 878 34 876 34 873 34 872 34 47 877 38 879 ...
31 :1182 16 942 18 15 922 19 18 16 922 16 18 15 922 16 18 18922 15 922 37 16 922 36 15 35 35 27922 34949 981 1021 35 940 34 937 35 935 34 932 35 933 ...
32 :1252 16 1005 19 16 984 20 18 16 984 17 19 16 984 16 18 18984 16 19966 37 16 984 38 16 74 27879 99 36 1011 1045 1088 1050 36 1001 35 998 36 996 35 995 36 ...
33 :1324 17 1070 19 16 1048 21 19 17 1048 18 19 16 1048 17 19 181048 16 19 102939 17 1048 38 16 108940 39 55 37 1075 371075 2276 36 1066 37 1064 36 1061 37 1059 ...
34 :1398 17 113720 17 1114 22 19 17 1114 19 20 17 1114 17 19 181114 17 20 109439 17 201094 40 17 1114 39 17 38 39 1141 38 1141 1179 1224 38 1134 37 1131 38 1129 37 ...
35 :1474 18 120620 17 1182 23 20 18 1182 20 20 17 1182 18 20 181182 17 20 1201 18 20 1161 40 17 1182 41 18 83 1095 107 40 1209 1248 1297 1256 39 41 1159 38 1198 39 1196 ...
36 :1552 18 1297 18 1252 24 20 18 1252 21 21 18 1252 19 20 181252 18 21 1270 18 20 1230 42 18 1252 41 18 1252 42 58 41 1279 421279 2701 39 41 1228 40 421228 2019 ...
37 :1632 19 1370 18 1324 25 21 19 1324 22 21 18 1324 20 21 191324 18 21 1342 19 21 1301 42 18 231301 43 19 1324 42 18 41 43 1351 43 1351 1395 144541 1346 40 42 ...
38 :1714 19 1445 19 1398 26 21 19 1398 23 22 19 1398 21 21 191398 19 22 1416 19 21 137444 19 23 1374 43 19 1398 44 19 91 1302 115 45 1425 441426 3004 42 43 1372 2219 ...
39 :1798 20 1522 19 1474 2722 20 1474 24 22 19 1474 22 22 201474 19 22 1492 20 22 144944 19 23 1449 45 20 251449 44 19 1474 45 62 46 1501 46 1502 316242 ...
40 :1884 20 1601 20 1552 2822 20 1552 25 23 20 1552 23 22 201552 20 23 1570 20 22 1571 20 23 1526 45 20 25 1526 46 20 1552 45 2044 48 1579 47 1579 1629 ...
41 :1972 21 1682 20 1632 51 21 1632 26 23 20 1632 24 23 211632 21 23 1650 21 23 1650 20 23 1605 47 21 25 1605 46 20 1632 47 21100 1527 123 49 1659 49...
42 :2062 21 1765 21 1714 52 21 1714 27 24 21 1714 25 23 211714 22 24 1733 21 23 1732 21 24 1686 47 21 25 1686 48 21 281686 47 21 1714 48 65 50...
43 :2154 22 1850 21 1798 53 22 1798 28 24 21 1798 26 24 221798 23 24 1817 22 24 1816 21 24 176949 22 25 1769 48 21 28 1769 49 22 1798 48 67 ...
44 :2248 22 1937 22 1884 54 22 1884 2925 22 1884 27 24 221884 24 25 1904 22 24 1902 22 25 185449 22 25 1854 50 22 28 1854 49 22 301854 50 ...
45 :2344 23 2026 22 1972 55 23 1972 3025 22 1972 28 25 231972 25 25 1992 23 25 1990 22 25 1991 23 25 1941 50 22 28 1941 51 23 30 1941 50 ...
46 :2442 23 2117 23 2062 56 23 2062 56 23 2062 29 25 232062 26 26 2083 24 25 2080 23 26 2080 23 25 2030 52 23 28 2030 51 23 30 2030 ...
47 :2542 24 2210 23 2154 57 24 2154 57 23 2154 30 26 242154 27 26 2175 25 26 2173 23 26 2172 24 26 2121 52 23 28 2121 53 24 30 ...
48 :2644 24 2305 24 2248 58 24 2248 58 24 2248 31 26 242248 28 27 2270 26 26 2267 24 27 2266 24 26 221454 24 28 2214 53 ...
49 :2748 25 2402 24 2344 59 25 2344 59 24 2344 3227 252344 29 27 2366 27 27 2364 24 27 2362 25 27 230954 24 28 2309 55 ...
50 :2854 25 292472 25 2442 60 25 2442 60 25 2442 3327 252442 30 28 2465 28 27 2462 25 28 2460 25 27 2461 25 28 2406 55 ...
51 :2962 26 302572 25 2542 61 26 2542 61 25 2542 61 262542 31 28 2565 29 28 2563 26 28 2560 26 28 2560 25 28 2505 57 ...
52 :3072 26 30 2674 26 2644 62 26 2644 62 26 2644 62 262644 32 29 2668 30 28 2665 27 29 2663 26 28 2662 26 29 ...
53 :3184 27 31 2778 26 2748 63 27 2748 63 26 2748 63 272748 33 29 2772 31 29 2770 28 29 2767 27 29 2766 26 29 ...
54 :3298 27 31 2884 27 2854 64 27 2854 64 27 2854 64 272854 3430 2879 32 29 2876 29 30 2874 27 29 2872 27 30 ...
55 :3414 28 32 2992 27 342928 65 28 2962 65 27 2962 65 282962 3530 2987 33 30 2985 30 30 2982 28 30 2980 27 ...
56 :3532 28 32 3102 28 353037 66 28 3072 66 28 3072 66 283072 66 3098 34 30 3095 31 31 3093 29 30 3090 28 ...
57 :3652 29 33 3214 28 35 3148 67 29 3184 67 28 3184 67 293184 67 3210 35 31 3208 32 31 3205 30 31 3203 ...
58 :3774 29 33 3328 29 36 3261 68 29 3298 68 29 3298 68 293298 68 3325 36 31 3322 33 32 3320 31 31 3317 ...
59 :3898 30 34 3444 29 36 3376 69 30 3414 69 29 3414 69 303414 69 3441 3732 3439 34 32 3436 32 32 3434 ...
60 :4024 30 34 3562 30 37 3493 70 30 393493 70 30 3532 70 303532 70 3560 3832 3557 35 33 3555 33 32 3552 ...
61 :4152 31 35 3682 30 37 3612 71 31 403612 71 30 3652 71 313652 71 3680 71 3678 36 33 3675 34 33 3673 ...
62 :4282 31 35 3804 31 38 3733 72 31 40 3733 72 31 3774 72 313774 72 3803 72 3800 37 34 3798 35 33 ...
63 :4414 32 36 3928 31 38 3856 73 32 41 3856 73 31 3898 73 323898 73 3927 73 3925 38 34 3922 36 34 ...
64 :4548 32 36 4054 32 39 3981 74 32 41 3981 74 32 4024 74 324024 74 4054 74 4051 3935 4049 37 34 ...
65 :4684 33 37 4182 32 39 4108 75 33 42 4108 75 32 444108 75 334152 75 4182 75 4180 4035 4177 38 35 ...
66 :4822 33 37 4312 33 40 4237 76 33 42 4237 76 33 454237 76 334282 76 4313 76 4310 76 4308 39 ...
67 :4962 34 38 4444 33 40 4368 77 34 43 4368 77 33 45 4368 77 344414 77 4445 77 4443 77 4440 ...
68 :5104 34 38 4578 34 41 4501 78 34 43 4501 78 34 46 4501 78 344548 78 4580 78 4577 78 4575 ...
69 :5248 35 39 4714 34 41 4636 79 35 44 4636 79 34 46 4636 79 354684 79 4716 79 4714 79 4711...
70 :5394 35 39 4852 35 42 4773 80 35 44 4773 80 35 47 4773 80 35494773 80 4855 80 4852 80 ...
71 :5542 36 40 4992 35 42 4912 81 36 45 4912 81 35 47 4912 81 36504912 81 4995 81 4993 81 ...
72 :5692 36 40 5134 36 43 5053 82 36 45 5053 82 36 48 5053 82 3650 5053 82 5138 82 5135 82 ...
73 :5844 37 41 5278 36 43 5196 83 37 46 5196 83 36 48 5196 83 3751 5196 83 5282 83 5280 83...
74 :5998 37 41 5424 37 44 5341 84 37 46 5341 84 37 49 5341 84 3751 5341 84 5429 84 5426 ...
75 :6154 38 42 5572 37 44 5488 85 38 47 5488 85 37 49 5488 85 3852 5488 85 895488 85 5575 ...
76 :6312 38 42 5722 38 45 5637 86 38 47 5637 86 38 50 5637 86 3852 5637 86 915637 86 ...
77 :6472 39 43 5874 38 45 5788 87 39 48 5788 87 38 50 5788 87 3953 5788 87 91 5788 87 ...
78 :6634 39 43 6028 39 46 5941 88 39 48 5941 88 39 51 5941 88 3953 5941 88 93 5941 88 ...
79 :6798 40 44 6184 39 46 6096 89 40 49 6096 89 39 51 6096 89 4054 6096 89 93 6096 89 ...
80 :6964 40 44 6342 40 47 6253 90 40 49 6253 90 40 52 6253 90 4054 6253 90 95 6253 ...
81 :7132 41 45 6502 40 47 6412 91 41 50 6412 91 40 52 6412 91 4155 6412 91 95 6412 ...
82 :7302 41 45 6664 41 48 6573 92 41 50 6573 92 41 53 6573 92 4155 6573 92 97 ...
83 :7474 42 46 6828 41 48 6736 93 42 51 6736 93 41 53 6736 93 4256 6736 93 ...
84 :7648 42 46 6994 42 49 6901 94 42 51 6901 94 42 54 6901 94 4256 6901 94 ...
85 :7824 43 47 7162 42 49 7068 95 43 52 7068 95 42 54 7068 95 4357 7068 95 ...
86 :8002 43 47 7332 43 50 7237 96 43 52 7237 96 43 55 7237 96 4357 7237 96 ...
87 :8182 44 48 7504 43 50 7408 97 44 53 7408 97 43 55 7408 97 4458 7408 97 ...
You can view more values Here, first $114$ number bases with all values each.
(zoom out to see more of the pattern at the same time)
How to generate the above pattern:
"Double" palindrome or Palindrome$^2$ - this is a number which is palindromic in two consecutive number bases $b,b+1$ . For example, number $10$ is a palindrome$^2$ (write $p^2$ short, from now on) since $10=22_3=101_4$
Now, consider only $p^2$ who have exactly $7$ digits in number bases $b,b+1$.
Each row starts with "$b:$" in the above pattern. Then every number after that in the row is the gap between two consecutive $p^2$'s divided by the smallest gap in the entire row (Which is $b+2$ in all observed rows so far); and ignore the remainder (round down).
All the smallest gaps divided by the smallest gap equal $1$. But you don't see any $1$'s in the pattern? That's because all $1$'s were replaced with an empty space "
".The gaps count the number of palindromes palindromic only in $b$, and not in $b+1$, which are in between two consecutive $p^2$'s.
How did I get this idea?
Well, I was trying to find a formula/algorithm to calculate/generate all $7$ digit $p^2$'s in a specific number base $b$. But I couldn't make sense of the patterns in the digits of consecutive $p^2$'s neither in their decimal values. Then I tried looking only at palindromes in base $b$ and at the gaps between two $p^2$'s. I noticed the smallest gap $(b+2)$ repeats a lot, so I divided every gap with the smallest gap while ignoring the remainder and replacing $1$'s with empty spaces. Then the pattern started to be more visible.
Question:
What is this pattern? Is there anything similar to it out there?
Q: How we determine/predict gaps at the next number base if we know the gaps at previous?
Observation: You can notice "splits" every $5$ number bases starting at $b=50$. In other words, a line of numbers splitting into two more lines. After the split, the line and lines around it seem to follow a constant pattern.
Example: notice 2402 at base $49$ becoming (splitting into) 29,2472 at base $50$.
Then 29 and 2472 start increasing by a constant pattern going down.
My python code for the above pattern: run it on repl.it!
How can we directly generate this pattern (exact numbers and gaps) without needing to check every single palindrome? (predicting the gaps/patterns?)
P.S. I managed to find a formula to generate all $p^2$'s with $3$ digits here, but it's rather straightforward unlike $7$ digits here. $9$ digits seem to have even more unpredictable gaps.
