Tuesday, February 18, 2020

The hidden palindromes


I submitted today, on behalf of Éric Angelini, OEIS A332661. The sequence (1, 2, 3, 11, 4, 22, 101, 5, 111, 6, 1001, 7, 88, 77, 8, 1111, 9, 10001, 33, 121, 131, 202, ...) is composed entirely of palindromic (base ten) integers with an added proviso that the products of adjacent terms are also palindromes. We don't see those products there, so I thought I'd show them here:

1. 1 * 2 = 2
2. 2 * 3 = 6
3. 3 * 11 = 33
4. 11 * 4 = 44
5. 4 * 22 = 88
6. 22 * 101 = 2222
7. 101 * 5 = 505
8. 5 * 111 = 555
9. 111 * 6 = 666
10. 6 * 1001 = 6006
11. 1001 * 7 = 7007
12. 7 * 88 = 616
13. 88 * 77 = 6776
14. 77 * 8 = 616
15. 8 * 1111 = 8888
16. 1111 * 9 = 9999
17. 9 * 10001 = 90009
18. 10001 * 33 = 330033
19. 33 * 121 = 3993
20. 121 * 131 = 15851
21. 131 * 202 = 26462
22. 202 * 44 = 8888
23. 44 * 2002 = 88088
24. 2002 * 141 = 282282
25. 141 * 212 = 29892
26. 212 * 1221 = 258852
27. 1221 * 303 = 369963
28. 303 * 2112 = 639936
29. 2112 * 222 = 468864
30. 222 * 3003 = 666666
31. 3003 * 232 = 696696
32. 232 * 10101 = 2343432
33. 10101 * 55 = 555555
34. 55 * 99 = 5445
35. 99 * 555 = 54945
36. 555 * 979 = 543345
37. 979 * 5555 = 5438345
38. 5555 * 9779 = 54322345
39. 9779 * 55555 = 543272345
40. 55555 * 97779 = 5432112345
41. 97779 * 100001 = 9777997779
42. 100001 * 66 = 6600066
43. 66 * 1000001 = 66000066
44. 1000001 * 151 = 151000151
45. 151 * 11011 = 1662661
46. 11011 * 161 = 1772771
47. 161 * 11111 = 1788871
48. 11111 * 171 = 1899981
49. 171 * 101101 = 17288271
50. 101101 * 181 = 18299281
51. 181 * 110011 = 19911991
52. 110011 * 242 = 26622662
53. 242 * 10201 = 2468642
54. 10201 * 313 = 3192913
55. 313 * 20002 = 6260626
56. 20002 * 323 = 6460646
57. 323 * 20102 = 6492946
58. 20102 * 333 = 6693966
59. 333 * 21012 = 6996996
60. 21012 * 404 = 8488848
61. 404 * 111111 = 44888844
62. 111111 * 252 = 27999972
63. 252 * 1001001 = 252252252
64. 1001001 * 191 = 191191191
65. 191 * 1010101 = 192929291
66. 1010101 * 262 = 264646462
67. 262 * 1100011 = 288202882
68. 1100011 * 272 = 299202992
69. 272 * 1101011 = 299474992
70. 1101011 * 343 = 377646773
71. 343 * 200002 = 68600686
72. 200002 * 414 = 82800828
73. 414 * 120021 = 49688694
74. 120021 * 1331 = 159747951
75. 1331 * 12021 = 15999951
76. 12021 * 10301 = 123828321
77. 10301 * 2222 = 22888822
78. 2222 * 20202 = 44888844
79. 20202 * 3113 = 62888826
80. 3113 * 11211 = 34899843
81. 11211 * 4004 = 44888844
82. 4004 * 210012 = 840888048
83. 210012 * 2332 = 489747984
84. 2332 * 30003 = 69966996
85. 30003 * 3223 = 96699669
86. 3223 * 300003 = 966909669
87. 300003 * 3333 = 999909999
88. 3333 * 1002001 = 3339669333
89. 1002001 * 424 = 424848424
90. 424 * 2000002 = 848000848
91. 2000002 * 434 = 868000868
92. 434 * 2001002 = 868434868
93. 2001002 * 444 = 888444888
94. 444 * 2002002 = 888888888
95. 2002002 * 1441 = 2884884882
96. 1441 * 201102 = 289787982
97. 201102 * 10401 = 2091661902
98. 10401 * 21112 = 219585912
99. 21112 * 11311 = 238797832
100. 11311 * 102201 = 1155995511
101. 102201 * 12121 = 1238778321
102. 12121 * 30103 = 364878463
103. 30103 * 12221 = 367888763
104. 12221 * 40004 = 488888884
105. 40004 * 112211 = 4488888844
106. 112211 * 31013 = 3479999743
107. 31013 * 1011101 = 31357275313
108. 1011101 * 4114 = 4159669514
109. 4114 * 1110111 = 4566996654
110. 1110111 * 10501 = 11657275611
111. 10501 * 1111111 = 11667776611
112. 1111111 * 10601 = 11778887711
113. 10601 * 10000001 = 106010010601
114. 10000001 * 282 = 2820000282
115. 282 * 10100101 = 2848228482
116. 10100101 * 292 = 2949229492
117. 292 * 100000001 = 29200000292
118. 100000001 * 353 = 35300000353
119. 353 * 10011001 = 3533883353
120. 10011001 * 363 = 3633993363
121. 363 * 11000011 = 3993003993
122. 11000011 * 454 = 4994004994
123. 454 * 11011011 = 4998998994
124. 11011011 * 2442 = 26888888862
125. 2442 * 20000002 = 48840004884
126. 20000002 * 3443 = 68860006886
127. 3443 * 20011002 = 68897879886
128. 20011002 * 11411 = 228345543822
129. 11411 * 1200021 = 13693439631
130. 1200021 * 12321 = 14785458741
131. 12321 * 1003001 = 12357975321
132. 1003001 * 20302 = 20362926302
133. 20302 * 121121 = 2458998542
134. 121121 * 301103 = 36469896463
135. 301103 * 21212 = 6386996836
136. 21212 * 310013 = 6575995756
137. 310013 * 1012101 = 313764467313
138. 1012101 * 13031 = 13188688131
139. 13031 * 2010102 = 26193639162
140. 2010102 * 12421 = 24967476942
141. 12421 * 2011102 = 24979897942
142. 2011102 * 22022 = 44288488244
143. 22022 * 400004 = 8808888088
144. 400004 * 22122 = 8848888488
145. 22122 * 1020201 = 22568886522
146. 1020201 * 32023 = 32669896623
147. 32023 * 1102011 = 35289698253
148. 1102011 * 22222 = 24488888442
149. 22222 * 2100012 = 46666466664
150. 2100012 * 13131 = 27575257572
151. 13131 * 2101012 = 27588388572
152. 2101012 * 13231 = 27798489772
153. 13231 * 3000003 = 39693039693
154. 3000003 * 13331 = 39993039993
155. 13331 * 10200201 = 135978879531
156. 10200201 * 13431 = 136998899631
157. 13431 * 20100102 = 269964469962
158. 20100102 * 21312 = 428373373824
159. 21312 * 2003002 = 42687978624
160. 2003002 * 30203 = 60496669406
161. 30203 * 1013101 = 30598689503
162. 1013101 * 30303 = 30699999603
163. 30303 * 3001003 = 90939393909
164. 3001003 * 22322 = 66988388966
165. 22322 * 10111101 = 225699996522
166. 10111101 * 10701 = 108198891801
167. 10701 * 11100111 = 118782287811
168. 11100111 * 4224 = 46886868864
169. 4224 * 100010001 = 422442244224
170. 100010001 * 373 = 37303730373
171. 373 * 100101001 = 37337673373
172. 100101001 * 383 = 38338683383
173. 383 * 101000101 = 38683038683
174. 101000101 * 393 = 39693039693
175. 393 * 101010101 = 39696969693
176. 101010101 * 464 = 46868686864
177. 464 * 1000000001 = 464000000464
178. 1000000001 * 474 = 474000000474
179. 474 * 1001001001 = 474474474474
180. 1001001001 * 484 = 484484484484
181. 484 * 1010000101 = 488840048884
182. 1010000101 * 494 = 498940049894
183. 494 * 10000000001 = 4940000000494
184. 10000000001 * 505 = 5050000000505
185. 505 * 110000011 = 55550005555
186. 110000011 * 515 = 56650005665
187. 515 * 110010011 = 56655155665
188. 110010011 * 525 = 57755255775
189. 525 * 1000110001 = 525057750525
190. 1000110001 * 535 = 535058850535
191. 535 * 1100000011 = 588500005885
192. 1100000011 * 545 = 599500005995
193. 545 * 923757329 = 503447744305
194. 923757329 * 5450000545 = 5034477946497744305
195. 5450000545 * 100000000001 = 545000054505450000545
196. 100000000001 * 565 = 56500000000565
197. 565 * 95359 = 53877835
198. 95359 * 565000565 = 53877888877835
199. 565000565 * 938839 = 530444565444035
200. 938839 * 565565 = 530974479035
201. 565565 * 1000000000001 = 565565000000565565
202. 1000000000001 * 575 = 575000000000575
203. 575 * 9119 = 5243425
204. 9119 * 575575 = 5248668425
205. 575575 * 911080119 = 524394939493425
206. 911080119 * 10000100001 = 9110892298922980119
207. 10000100001 * 585 = 5850058500585
208. 585 * 10010001001 = 5855850585585
209. 10010001001 * 595 = 5955950595595
210. 595 * 92829 = 55233255
211. 92829 * 54145 = 5026226205
212. 54145 * 9282992829 = 502627646726205
213. 9282992829 * 546808645 = 5076020730370206705
214. 546808645 * 10000000000001 = 5468086450000546808645
215. 10000000000001 * 606 = 6060000000000606
216. 606 * 1100110011 = 666666666666
217. 1100110011 * 616 = 677667766776
218. 616 * 11000000011 = 6776000006776
219. 11000000011 * 626 = 6886000006886
220. 626 * 11000100011 = 6886062606886
221. 11000100011 * 636 = 6996063606996
222. 636 * 100001100001 = 63600699600636
223. 100001100001 * 707 = 70700777700707
224. 707 * 858 = 606606
225. 858 * 777 = 666666
226. 777 * 858000858 = 666666666666
227. 858000858 * 70007 = 60066066066006
228. 70007 * 88088 = 6166776616
229. 88088 * 7000007 = 616616616616
230. 7000007 * 8580858 = 60066066066006
231. 8580858 * 700007 = 6006660666006
232. 700007 * 880088 = 616067760616
233. 880088 * 7007 = 6166776616
234. 7007 * 88000088 = 616616616616
235. 88000088 * 70000007 = 6160006776000616
236. 70000007 * 85800858 = 6006060660606006
237. 85800858 * 7000000007 = 600606006600606006
238. 7000000007 * 8800088 = 61600616061600616
239. 8800088 * 700000007 = 6160061661600616
240. 700000007 * 11111111 = 7777777777777777
241. 11111111 * 11511 = 127899998721
242. 11511 * 100111001 = 1152377732511
243. 100111001 * 10801 = 1081298921801
244. 10801 * 101101101 = 1091992991901
245. 101101101 * 11611 = 1173884883711
246. 11611 * 110101011 = 1278382838721
247. 110101011 * 11711 = 1289392939821
248. 11711 * 111000111 = 1299922299921
249. 111000111 * 5005 = 555555555555
250. 5005 * 1110000111 = 5555550555555
251. 1110000111 * 5115 = 5677650567765
252. 5115 * 10001010001 = 51155166155115
253. 10001010001 * 1551 = 15511566511551
254. 1551 * 10010101001 = 15525666652551
255. 10010101001 * 1661 = 16626777762661
256. 1661 * 10100000101 = 16776100167761
257. 10100000101 * 1771 = 17887100178871
258. 1771 * 10100100101 = 17887277278871
259. 10100100101 * 1881 = 18998288289981
260. 1881 * 10101010101 = 18999999999981
261. 10101010101 * 2552 = 25777777777752
262. 2552 * 100010010001 = 255225545522552
263. 100010010001 * 646 = 64606466460646
264. 646 * 100100001001 = 64664600646646
265. 100100001001 * 656 = 65665600656656
266. 656 * 1000001000001 = 656000656000656
267. 1000001000001 * 666 = 666000666000666
268. 666 * 1000100010001 = 666066606660666
269. 1000100010001 * 676 = 676067606760676
270. 676 * 1001000001001 = 676676000676676
271. 1001000001001 * 686 = 686686000686686
272. 686 * 1001001001001 = 686686686686686
273. 1001001001001 * 696 = 696696696696696
274. 696 * 10000100100001 = 6960069669600696
275. 10000100100001 * 717 = 7170071771700717
276. 717 * 874478 = 627000726
277. 874478 * 7887 = 6897007986
278. 7887 * 87470107478 = 689876737678986
279. 87470107478 * 100000000000001 = 8747010747800087470107478
280. 100000000000001 * 727 = 72700000000000727
281. 727 * 100110011001 = 72779977997727
282. 100110011001 * 808 = 80888888888808
283. 808 * 110000000011 = 88880000008888
284. 110000000011 * 818 = 89980000008998
285. 818 * 7580990857 = 6201250521026
286. 7580990857 * 8998 = 68213755731286
287. 8998 * 7587447857 = 68271855817286
288. 7587447857 * 1000000000000001 = 7587447857000007587447857
289. 1000000000000001 * 737 = 737000000000000737
290. 737 * 888 = 654456
291. 888 * 737000737 = 654456654456
292. 737000737 * 888000000000888 = 654456654456654456654456
293. 888000000000888 * 1000001001000001 = 888000888888888888888888000888
294. 1000001001000001 * 747 = 747000747747000747
295. 747 * 8274728 = 6181221816
296. 8274728 * 7470747 = 61818399381816
297. 7470747 * 82747255274728 = 618183809101908381816
298. 82747255274728 * 74700747 = 6181281781221871821816
299. 74700747 * 10000000000000001 = 747007470000000074700747
300. 10000000000000001 * 757 = 7570000000000000757
301. 757 * 8421248 = 6374884736
302. 8421248 * 75700000757 = 637488479974884736
303. 75700000757 * 100000000000000001 = 7570000075700000075700000757
304. 100000000000000001 * 767 = 76700000000000000767
305. 767 * 839938 = 644232446
306. 839938 * 767767 = 644876678446
307. 767767 * 100000010000001 = 76776707677670767767
308. 100000010000001 * 787 = 78700007870000787
309. 787 * 10001000010001 = 7870787007870787
310. 10001000010001 * 797 = 7970797007970797
311. 797 * 10010000001001 = 7977970000797797
312. 10010000001001 * 828 = 8288280000828828
313. 828 * 74275657247 = 61500244200516
314. 74275657247 * 828828 = 61561744444716516
315. 828828 * 7420880247 = 6150633333360516
316. 7420880247 * 828828828828 = 6150639483993849360516
317. 828828828828 * 1000000000000000001 = 828828828828000000828828828828
318. 1000000000000000001 * 838 = 838000000000000000838
319. 838 * 765567 = 641545146
320. 765567 * 838000838 = 641545787545146
321. 838000838 * 1001000000001001 = 838838838838838838838838
322. 1001000000001001 * 848 = 848848000000848848
323. 848 * 100001000100001 = 84800848084800848
324. 100001000100001 * 868 = 86800868086800868
325. 868 * 79901510997 = 69354511545396
326. 79901510997 * 86800000000000868 = 6935451154539669354511545396
327. 86800000000000868 * 100000010010000001 = 8680000868868086886808688680000868
328. 100000010010000001 * 878 = 87800008788780000878
329. 878 * 100010000010001 = 87808780008780878
330. 100010000010001 * 898 = 89808980008980898
331. 898 * 7227 = 6489846
332. 7227 * 8980000898 = 64898466489846
333. 8980000898 * 10010000000001001 = 89889808988988988980898898
334. 10010000000001001 * 909 = 9099090000000909909
335. 909 * 110001100011 = 99990999909999
336. 110001100011 * 4334 = 476744767447674
337. 4334 * 100020001 = 433486684334
338. 100020001 * 4444 = 444488884444
339. 4444 * 200000002 = 888800008888
340. 200000002 * 14041 = 2808200028082
341. 14041 * 12000021 = 168492294861
342. 12000021 * 14141 = 169692296961
343. 14141 * 21000012 = 296961169692
344. 21000012 * 20402 = 428442244824
345. 20402 * 220022 = 4488888844
346. 220022 * 202202 = 44488888444
347. 202202 * 1021201 = 206488884602
348. 1021201 * 130031 = 132787787231
349. 130031 * 1201021 = 156169961651
350. 1201021 * 122221 = 146789987641
351. 122221 * 2110112 = 257899998752
352. 2110112 * 103301 = 217976679712
353. 103301 * 2020202 = 208688886802
354. 2020202 * 211112 = 426488884624
355. 211112 * 1004001 = 211956659112
356. 1004001 * 1120211 = 1124692964211
357. 1120211 * 1112111 = 1245798975421
358. 1112111 * 1210121 = 1345788875431
359. 1210121 * 1022201 = 1236986896321
360. 1022201 * 1103011 = 1127498947211
361. 1103011 * 221122 = 243899998342
362. 221122 * 3010103 = 665599995566
363. 3010103 * 212212 = 638779977836
364. 212212 * 3100013 = 657859958756
365. 3100013 * 1030301 = 3193946493913
366. 1030301 * 2200022 = 2266684866622
367. 2200022 * 1031301 = 2268884888622
368. 1031301 * 3002003 = 3095968695903
369. 3002003 * 1121211 = 3365878785633
370. 1121211 * 2102012 = 2356798976532
371. 2102012 * 31113 = 65399899356
372. 31113 * 12011021 = 373698896373
373. 12011021 * 31213 = 374899998473
374. 31213 * 21100112 = 658597795856
375. 21100112 * 23032 = 485977779584
376. 23032 * 12100121 = 278689986872
377. 12100121 * 23132 = 279899998972
378. 23132 * 30000003 = 693960069396
379. 30000003 * 23232 = 696960069696
380. 23232 * 100030001 = 2323896983232
381. 100030001 * 23332 = 2333899983332
382. 23332 * 102000201 = 2379868689732
383. 102000201 * 104401 = 10648922984601
384. 104401 * 101020101 = 10546599564501
385. 101020101 * 33033 = 3336996996333
386. 33033 * 111010111 = 3666996996663
387. 111010111 * 12521 = 1389957599831
388. 12521 * 110111011 = 1378699968731
389. 110111011 * 20502 = 2257495947522
390. 20502 * 101111101 = 2072979792702
391. 101111101 * 40104 = 4054959594504
392. 40104 * 22000022 = 882288882288
393. 22000022 * 30403 = 668866668866
394. 30403 * 10022001 = 304698896403
395. 10022001 * 302203 = 3028678768203
396. 302203 * 31000013 = 9368296928639
397. 31000013 * 222222 = 6888884888886
398. 222222 * 4000004 = 888888888888
399. 4000004 * 1122211 = 4488848488844
400. 1122211 * 11200211 = 12568999986521
401. 11200211 * 1202021 = 13462888826431
402. 1202021 * 2012102 = 2418588858142
403. 2012102 * 410014 = 824989989428
404. 410014 * 110020011 = 45109744790154
405. 110020011 * 14241 = 1566794976651
406. 14241 * 200010002 = 2848342438482
407. 200010002 * 14341 = 2868343438682
408. 14341 * 200020002 = 2868486848682
409. 200020002 * 14441 = 2888488848882
410. 14441 * 200101002 = 2889658569882
411. 200101002 * 21412 = 4284562654824
412. 21412 * 21011012 = 449887788944
413. 21011012 * 311113 = 6536798976356
414. 311113 * 100121001 = 31148944984113
415. 100121001 * 131131 = 13128966982131
416. 131131 * 100202001 = 13139588593131
417. 100202001 * 32123 = 3218788878123
418. 32123 * 120000021 = 3854760674583
419. 120000021 * 31313 = 3757560657573
420. 31313 * 120010021 = 3757873787573
421. 120010021 * 31413 = 3769874789673
422. 31413 * 210000012 = 6596730376956
423. 210000012 * 32223 = 6766830386676
424. 32223 * 121000121 = 3898986898983
425. 121000121 * 40204 = 4864688864684
426. 40204 * 220000022 = 8844880884488
427. 220000022 * 40304 = 8866880886688
428. 40304 * 1001111001 = 40348777784304
429. 1001111001 * 20602 = 20624888842602
430. 20602 * 1011001101 = 20828644682802
431. 1011001101 * 12621 = 12759844895721
432. 12621 * 1010110101 = 12748599584721
433. 1010110101 * 13531 = 13667799776631
434. 13531 * 1101001011 = 14897644679841
435. 1101001011 * 20702 = 22792922929722
436. 20702 * 1111001111 = 22999944999922
437. 1111001111 * 21512 = 23899855899832
438. 21512 * 10001110001 = 215143878341512
439. 10001110001 * 5225 = 52255799755225
440. 5225 * 11100000111 = 57997500579975
441. 11100000111 * 6006 = 66666600666666
442. 6006 * 101000000101 = 606606000606606
443. 101000000101 * 2662 = 268862000268862
444. 2662 * 101010010101 = 268888646888862
445. 101010010101 * 2772 = 279999747999972
446. 2772 * 1000010100001 = 2772027997202772
447. 1000010100001 * 3553 = 3553035885303553
448. 3553 * 1000101010001 = 3553358888533553
449. 1000101010001 * 3663 = 3663369999633663
450. 3663 * 1001010101001 = 3666699999966663
451. 1001010101001 * 4554 = 4558599999958554
452. 4554 * 1010000000101 = 4599540000459954
453. 1010000000101 * 5335 = 5388350000538835
454. 5335 * 969 = 5169615
455. 969 * 533505335 = 516966669615
456. 533505335 * 96769 = 51626777762615
457. 96769 * 58685 = 5678888765
458. 58685 * 9676769 = 567881188765
459. 9676769 * 53350533505335 = 516260788757887062615
460. 53350533505335 * 967676769 = 51626071886868817062615

#303 to #460 were added on 29 April 2022.

Note that there are a few duplicate products. I count 449 distinct entries, ten of which have duplicates: nine with one duplicate and one with two duplicates.

616: 2
8888: 2
666666: 2
44888844: 3
44888888442
6166776616: 2
39693039693: 2
616616616616: 2
666666666666: 2
60066066066006: 2

I have a new picture of the now-461 sequence terms in the updated OEIS.

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