On Echo Beach, waves make the only sound. On Echo Beach, there's not a soul around.
I am sitting on the back deck in the glorious sunshine, listening to some random tunes in my Join-In-Daily playlist. Could life be any sweeter?
Thursday, April 2
Echo Beach
On Echo Beach, waves make the only sound. On Echo Beach, there's not a soul around.
I am sitting on the back deck in the glorious sunshine, listening to some random tunes in my Join-In-Daily playlist. Could life be any sweeter?
Saturday, March 28
Thursday, March 26
Bare necessities
Our last grocery shop was on March 11 at the local Real Canadian Superstore. We only go every three or four weeks so it's important to stock up. Alas, already back then they had no bathroom tissue to sell. So on March 17 I decided to try a grocery delivery service, more specifically Grocery Gateway. There's a minimum $50 order so in addition to $30 worth of "Cashmere" I ordered some canned goods. At checkout, the website wouldn't recognize my credit card information so I opted to pay at the door. I tried subsequently to add my credit card information to the account but there was no way to do that. I'm still waiting for their email response to my query about it. But no matter, when the order arrives I'll tap the credit card so that I won't have to push the buttons.
The scheduled arrival for the order was this morning. I was waiting for it by the front steps. When the order was brought to me I asked about the tap limit. I think he said $50. Damn! My original order was for just over $80. But wait, where's the toilet paper? They didn't include it, which put my total owing under $50. But my tap didn't work for some reason so I had to push the buttons.
The scheduled arrival for the order was this morning. I was waiting for it by the front steps. When the order was brought to me I asked about the tap limit. I think he said $50. Damn! My original order was for just over $80. But wait, where's the toilet paper? They didn't include it, which put my total owing under $50. But my tap didn't work for some reason so I had to push the buttons.
Sunday, March 22
My 500th Leyland prime find
This morning, after a four-and-a-half-day wait, I found my 500th and 501st Leyland primes. The above graph extends what I showed for my 200th find. I have now surpassed Anatoly Selevich's 475 such finds that he computed from January 2003 to July 2011.
Generally, I'm happy with the ongoing search. My 54 dedicated Mac-mini cores have been supplemented in the last few months by 6 cores on my old Mac Pro and 4 on a more recent iMac, which have been working on interval #8 to gain time on the overall computation, the length of which I now realize I did not correctly calculate. More specifically, the three Mac minis that have been working the upper half of interval #14 since early October 2019 were thought to complete their task by July of this year. Instead, they will run for a full year, until October 2020. In effect, that pushes the overall expected spring-2021 completion date to the fall of that year.
Thursday, March 5
Primes describing digit position
On Monday, Éric Angelini posted this to the Sequence Fanatics Discussion list: S = 11, 41, 61, 83, 113, 101, ... with digits 1, 1, 4, 1, 6, 1, 8, 3, 1, 1, 3, 1, 0, 1, ... at positions 1, 2, 3, 4, ...
11 says: "In position 1 is a 1."
41 says: "In position 4 is a 1."
61 says: "In position 6 is a 1."
83 says: "In position 8 is a 3."
113 says: "In position 11 is a 3."
101 says: "In position 10 is a 1."
etc.
Of course, each added prime must be the smallest possible that has not already been used. There's a few early surprises hinting at things to come: 11, 41, 61, 83, 113, 101, 151, 181, 233, 223, 263, 293, 353, 383, 419, 401, 479, 467, 541, 1009, 599, 631, 661, 691, 727, 751, 787, 797, 809, 877, 907, 919, 967, 991, 9001, 1031, ... Term #20 is 1009 because to the end of term #19 we have 53 digits/positions and term #19 says that the next digit (position 54) is a 1. So we need a prime starting with 1 and 1009 is the smallest one that keeps the growing sequence truthful. Term #20 also dictates that in position 100 is a 9. So when we get to term #34 = 991, we now have 99 digits/positions and so the next prime must start with a 9. Why not 997? Because that says that in position 99 is a 7 and we already know that in position 99 is a 1. So we must travel all the way up to 9001 to keep things honest. And that may have repercussions when we get to position 900.
I eventually wrote a Mathematica program that seemed to work extending the sequence. But it was taking a long time finding term #1447. So I had a look at how far it had gotten. Term #1446 was 190901 taking up positions 7006-7011. Perusing the list of prior terms, I saw that positions 7012-7020 and 7022-7024 were already assigned with digits: 191737191?371... Stepping through, 19 is prime, as is 191, but these lie: position 1 is not 9; position 19 is not 1. Continuing, no more primes up to 191737191. Then we can try 1917371911, 1917371913, 1917371917, 1917371919, replacing the ? with 1, 3, 7, 9, but these are not prime either. So we attach the next digit, 3, and replace the ? with 0, 1, 2, 3, ..., 9. We need not go further than 5 because 19173719153, finally, is prime!
So I managed to figure out term #1447 before my program did! In fact, it would not have found it because I had my initial search go up to only 104395301. Here's a graph (click on it) of 1500 terms:
11 says: "In position 1 is a 1."
41 says: "In position 4 is a 1."
61 says: "In position 6 is a 1."
83 says: "In position 8 is a 3."
113 says: "In position 11 is a 3."
101 says: "In position 10 is a 1."
etc.
Of course, each added prime must be the smallest possible that has not already been used. There's a few early surprises hinting at things to come: 11, 41, 61, 83, 113, 101, 151, 181, 233, 223, 263, 293, 353, 383, 419, 401, 479, 467, 541, 1009, 599, 631, 661, 691, 727, 751, 787, 797, 809, 877, 907, 919, 967, 991, 9001, 1031, ... Term #20 is 1009 because to the end of term #19 we have 53 digits/positions and term #19 says that the next digit (position 54) is a 1. So we need a prime starting with 1 and 1009 is the smallest one that keeps the growing sequence truthful. Term #20 also dictates that in position 100 is a 9. So when we get to term #34 = 991, we now have 99 digits/positions and so the next prime must start with a 9. Why not 997? Because that says that in position 99 is a 7 and we already know that in position 99 is a 1. So we must travel all the way up to 9001 to keep things honest. And that may have repercussions when we get to position 900.
I eventually wrote a Mathematica program that seemed to work extending the sequence. But it was taking a long time finding term #1447. So I had a look at how far it had gotten. Term #1446 was 190901 taking up positions 7006-7011. Perusing the list of prior terms, I saw that positions 7012-7020 and 7022-7024 were already assigned with digits: 191737191?371... Stepping through, 19 is prime, as is 191, but these lie: position 1 is not 9; position 19 is not 1. Continuing, no more primes up to 191737191. Then we can try 1917371911, 1917371913, 1917371917, 1917371919, replacing the ? with 1, 3, 7, 9, but these are not prime either. So we attach the next digit, 3, and replace the ? with 0, 1, 2, 3, ..., 9. We need not go further than 5 because 19173719153, finally, is prime!
So I managed to figure out term #1447 before my program did! In fact, it would not have found it because I had my initial search go up to only 104395301. Here's a graph (click on it) of 1500 terms:
Updates:
Sunday, March 8: I have run into a second large term at #3868. Term #3867 was 301471 taking up positions 21005-21010. Positions 21011-21020 and 21022-21028 were already assigned with digits: 3713793719?9317373... So #3868 is 371379371929 and #3869 is 31737313.
Monday, March 9: This is now OEIS A333085. It needs a decent Mathematica program!
Wednesday, March 11: I have rewritten my original program to run significantly faster. In fact, the new version has already overtaken the number of terms calculated by the old one. Here's an updated graph.
Monday, March 16: I've reached 12000 terms and primes strictly greater than 700000.
Thursday, March 19: Maximilian Hasler has written PARI/GP code for this sequence which he says computes 10000 terms in a few seconds. Unfortunately, I haven't been able to get it to run.
Tuesday, March 31: I've decided to call it quits at 18000 terms. There was another spike at #16966.
Friday, February 28
Transcription
Yesterday, Futility Closet showcased a nice arithmetical/linguistic limerick that I hadn't previously seen, sourcing it to "Punch, Sept. 29, 1937, via William R. Ransom, One Hundred Mathematical Curiosities, 1953". The Ransom work appears to be typewritten, as evidenced by a Google snippet view from the 1955 book (I think Futility Closet, via Google perhaps, misstated the year):
Already we see some differences from the limerick in the Futility Closet version. Two lines are indented; there are spaces on either side of the equals sign; most importantly, 3/8 actually is 3 over 8. Interest thus whetted, of course we'd like to see the original from Punch:
Oh my! It's written as a sentence, so no capitalizations at the start of the introduced lines; Ransom's 'Maths Master' also is not capitalized; the period after π is a comma; there's a comma after the fraction (to introduce a pause, not necessary in the lines version); the equals sign has no spaces on its sides, but it is extra long (though not as long as Robert Recorde's 1557 original); and the finale: there is no 'why'.
Already we see some differences from the limerick in the Futility Closet version. Two lines are indented; there are spaces on either side of the equals sign; most importantly, 3/8 actually is 3 over 8. Interest thus whetted, of course we'd like to see the original from Punch:
Oh my! It's written as a sentence, so no capitalizations at the start of the introduced lines; Ransom's 'Maths Master' also is not capitalized; the period after π is a comma; there's a comma after the fraction (to introduce a pause, not necessary in the lines version); the equals sign has no spaces on its sides, but it is extra long (though not as long as Robert Recorde's 1557 original); and the finale: there is no 'why'.
Tuesday, February 18
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
Note that there are a few duplicate products. I count 292 distinct entries, nine of which have duplicates: eight with one duplicate and one with two duplicates.
616: 2
8888: 2
666666: 2
44888844: 3
6166776616: 2
39693039693: 2
616616616616: 2
666666666666: 2
60066066066006: 2
Friday, February 14
What's so special about these numbers?
By which I mean the numbers in the first column. Each of those numbers is followed by the product of the number's digits, which is followed by the number's square, which is followed by the number's cube.
0 0 0 0
1 1 1 1
93 27 8649 804357
1941 36 3767481 7312680621
22911 36 524913921 12026302844031
23313 54 543495969 12670521525297
111633 54 12461926689 1391162262073137
113163 54 12805864569 1449150052221747
115911 45 13435359921 1557306003803031
136311 54 18580688721 2532752260248231
192114 72 36907788996 7090502975177544
631122 72 398314978884 251385346103227848
631131 54 398326339161 251396100761021091
631212 72 398428588944 251492906484520128
912114 72 831951948996 758835020006537544
1111197 63 1234758772809 1372060244069042373
1111917 63 1236359414889 1374729051525132213
1312911 54 1723735293921 2263111028477114031
1334121 72 1779878842641 2374573741423053561
1831311 72 3353699978721 6141667661731533231
3111423 72 9680953084929 30121540090369043967
3113313 81 9692717835969 30176464444054155297
3114312 72 9698939233344 30205522841674019328
3121134 72 9741477445956 30404456466806434104
3143211 72 9879775390521 31054218685014902931
9211212 72 84846426508944 781538422016303080128
11213232 72 125736571885824 1409913351440422023168
13212231 72 174563047997361 2306367314205220922391
16111341 72 259575308818281 4182106315551632224821
22311312 72 497794643161344 11106451595501412323328
23221311 72 539229284558721 12521610917045558103231
111142911 72 12352746665553921 1372920223255206207404031
111632121 72 12461730438958641 1391129400231214126107561
131135121 90 17196419959684641 2255054612180060519376561
132113511 90 17453979788747121 2305906550814420446451831
132135111 90 17459687558982321 2307037753631448032372631
151151212 100 22846688889068944 3453304715769704437160128
211151125 100 44584797588765625 9414130168765149080078125
211511313 90 44737035526983969 9462389124040026213141297
311123151 90 96797615088168801 30115979015516220187011951
311139111 81 96807546393870321 30120613923080067525224631
311262111 72 96884101744176321 30156350031231103630673631
311314113 108 96916476952976769 30171467057700905550840897
312111513 90 97413596547149169 30403905005102302973282697
312151131 90 97438328584579161 30415484470426014265181091
313511211 90 98289279422686521 30814791020123832072086931
313512111 90 98289843743676321 30815056401940106297423631
314111313 108 98665916954583969 30992080722953331871341297
353111121 90 124687463773876641 44028530107840471219224561
521111215 100 271556898398776225 141511345266217833122863375
921211131 108 848629947878299161 781767354085439020461161091
1132112133 108 1281677881685809689 1451003080454243642845856637
1132121313 108 1281698667348843969 1451038378149323463216411297
1133123112 108 1283967986948564544 1454893801079532840030140928
1231132113 108 1515686279659844769 1866010052122733611710966897
1311331212 108 1719589547565388944 2254951445551453133186920128
1321211313 108 1745599333599183969 2306305587516502867411041297
1322111313 108 1747978323962583969 2311021916989711254127341297
1651111131 90 2726167966912099161 4501206275144206623302861091
1961111121 108 3845956828909876641 7542348708061053387403224561
2121113313 108 4499121686585835969 9543146906224230191080155297
2211311331 108 4889897802608991561 10813086418341264401322677691
2211331131 108 4889985370929739161 10813376880871514620429121091
2231131311 108 4977946926924578721 11106453253157656519907433231
3111122133 108 9679080926442469689 30112802897353312400629526637
3111123312 108 9679088262469849344 30112837132275522991246307328
3111223113 108 9679709258865410769 30115735173302166140751903897
3113131212 108 9691585943128588944 30171178693334067171084520128
3311111322 108 10963458186676587684 36301230530178439032818158248
6911111121 108 47763456926809876641 330098558344280221506253224561
11112136131 108 123479569393875649161 1372121784502007371163027936091
11116321311 108 123572599489392758721 1373672721159604442219333403231
11133611121 108 123957296593654876641 1380092335884211352606120724561
11213161131 108 125734982549769199161 1409886619134035257053123011091
11311321611 108 127945996587475635321 1447238316240845405742382222131
11312161131 108 127964989453707199161 1447560579827051502203940011091
11613111117 126 134864349815788987689 1566194680132715995057302038613
21111316311 108 445687676383094648721 9409053512038115542200462588231
21113611311 108 445784582591987138721 9412122405283593350066231673231
26311111131 108 692274568947832099161 18214513116751532102554082861091
31111119141 108 967901734205496577881 30112506169647719011523576320221
31111121163 108 967901860018866472569 30112512040940020294012574877747
31111613211 108 967932476590869730521 30113940826060450950057153512931
31111711116 126 967938568565437965456 30114225123242464523285859208896
31114141113 144 968089777199676878769 30121281937843476384022059729897
31141141113 144 969770669819778878769 30199765276106264342202348729897
61112131311 108 3734692593372906578721 228235024172424195228533510433231
71311161111 126 5085281698998998754321 362637342532137407597073138410631
111121142313 144 12347908268945998989969 1372113672021417832276299508458297
112119111117 126 12570695077666192987689 1409415158230780836621069204038613
112143211113 144 12576099798734886698769 1410324214707683209378187604219897
112191112311 108 12586845681579415760721 1412132217503301577476813213336231
119121111117 126 14189839113748660987689 1690309401801207047924464338038613
121131211911 108 14672770499027588271921 1777330472639180014401320942051031
129111112131 108 16669679275703655361161 2152240830153181511261437173344091
132291111111 108 17500938078982947654321 2315218543953464022012076230260631
134111221113 144 17985819628419976958769 2412100233085567028743834643289897
141211112133 144 19940578189838699809689 2815831222762166800008919138856637
161361111111 108 26037408178976487654321 4201425114210285198645410640260631
161613111111 108 26118797682976431654321 4221140152024599402144077316260631
211111312911 108 44567986439006155293921 9408806130938233066110015407114031
211132113114 144 44576769187982890776996 9411587474453873625025426821125544
213211111134 144 45458977910994898765956 9692359191419184521295324171754104
221111113911 108 48890124694963217715921 10810149930551006160340101469277031
221111191131 108 48890158843369613059161 10810161256441248405201661481501091
221131111413 144 48898968434748618856569 10813083236925167057019832205921997
221131131114 144 48898977147757058880996 10813086127001155938432544598909544
221311111911 108 48978608255283166071921 10839510252830001223537306705751031
223111411113 144 49778701768834099898769 11106196395017765153207562041619897
231211141113 144 53458591774775598878769 12360222006539902106268143838729897
292131111111 108 85340586078947427654321 24930640234106850806012082150260631
311111133141 108 96790137164277028525881 30112489250051042810723218755322221
311111213214 144 96790186987486968209796 30112512500886986516764283239444344
311111314131 108 96790249780317760285161 30112541804222392461075080008910091
311113141122 144 96791386578797487418884 30113072312083479476410277419747848
311113411122 144 96791554579966593298884 30113150713174648766021468115787848
311114112312 144 96791990879683749985344 30113354321444009871981641569955328
311121241311 144 96796426794897492998721 30115424458897849222764009255363231
311212114113 144 96852979970682933776769 30141820654820280581077456096440897
311212141311 144 96852996899377832798721 30141828557443018948936490472063231
311221131114 144 96858592451877578880996 30144440700983282712399333518909544
311221141131 144 96858598686881819959161 30144443611680938167828178237351091
312211111911 108 97475778400702966071921 30433021158873710341350163405751031
312211411113 144 97475965229170699898769 30433108653801106646148515741619897
312341111112 144 97556969690678729876544 30471052309906301439064663546556928
314111112231 144 98665790826995877797361 30992021295818872475381222146622391
321121111911 108 103118768514956986071921 33113613604416005601752611735751031
332111112114 144 110297790789597877548996 36631121962850657295651054084137544
431111113212 144 185856791934889880956944 80124928469061440001982287681544128
431111121132 144 185856798763689976961424 80124932885018897501110751355211968
Partial spoiler: The numbers are a subset of A117224.
0 0 0 0
1 1 1 1
93 27 8649 804357
1941 36 3767481 7312680621
22911 36 524913921 12026302844031
23313 54 543495969 12670521525297
111633 54 12461926689 1391162262073137
113163 54 12805864569 1449150052221747
115911 45 13435359921 1557306003803031
136311 54 18580688721 2532752260248231
192114 72 36907788996 7090502975177544
631122 72 398314978884 251385346103227848
631131 54 398326339161 251396100761021091
631212 72 398428588944 251492906484520128
912114 72 831951948996 758835020006537544
1111197 63 1234758772809 1372060244069042373
1111917 63 1236359414889 1374729051525132213
1312911 54 1723735293921 2263111028477114031
1334121 72 1779878842641 2374573741423053561
1831311 72 3353699978721 6141667661731533231
3111423 72 9680953084929 30121540090369043967
3113313 81 9692717835969 30176464444054155297
3114312 72 9698939233344 30205522841674019328
3121134 72 9741477445956 30404456466806434104
3143211 72 9879775390521 31054218685014902931
9211212 72 84846426508944 781538422016303080128
11213232 72 125736571885824 1409913351440422023168
13212231 72 174563047997361 2306367314205220922391
16111341 72 259575308818281 4182106315551632224821
22311312 72 497794643161344 11106451595501412323328
23221311 72 539229284558721 12521610917045558103231
111142911 72 12352746665553921 1372920223255206207404031
111632121 72 12461730438958641 1391129400231214126107561
131135121 90 17196419959684641 2255054612180060519376561
132113511 90 17453979788747121 2305906550814420446451831
132135111 90 17459687558982321 2307037753631448032372631
151151212 100 22846688889068944 3453304715769704437160128
211151125 100 44584797588765625 9414130168765149080078125
211511313 90 44737035526983969 9462389124040026213141297
311123151 90 96797615088168801 30115979015516220187011951
311139111 81 96807546393870321 30120613923080067525224631
311262111 72 96884101744176321 30156350031231103630673631
311314113 108 96916476952976769 30171467057700905550840897
312111513 90 97413596547149169 30403905005102302973282697
312151131 90 97438328584579161 30415484470426014265181091
313511211 90 98289279422686521 30814791020123832072086931
313512111 90 98289843743676321 30815056401940106297423631
314111313 108 98665916954583969 30992080722953331871341297
353111121 90 124687463773876641 44028530107840471219224561
521111215 100 271556898398776225 141511345266217833122863375
921211131 108 848629947878299161 781767354085439020461161091
1132112133 108 1281677881685809689 1451003080454243642845856637
1132121313 108 1281698667348843969 1451038378149323463216411297
1133123112 108 1283967986948564544 1454893801079532840030140928
1231132113 108 1515686279659844769 1866010052122733611710966897
1311331212 108 1719589547565388944 2254951445551453133186920128
1321211313 108 1745599333599183969 2306305587516502867411041297
1322111313 108 1747978323962583969 2311021916989711254127341297
1651111131 90 2726167966912099161 4501206275144206623302861091
1961111121 108 3845956828909876641 7542348708061053387403224561
2121113313 108 4499121686585835969 9543146906224230191080155297
2211311331 108 4889897802608991561 10813086418341264401322677691
2211331131 108 4889985370929739161 10813376880871514620429121091
2231131311 108 4977946926924578721 11106453253157656519907433231
3111122133 108 9679080926442469689 30112802897353312400629526637
3111123312 108 9679088262469849344 30112837132275522991246307328
3111223113 108 9679709258865410769 30115735173302166140751903897
3113131212 108 9691585943128588944 30171178693334067171084520128
3311111322 108 10963458186676587684 36301230530178439032818158248
6911111121 108 47763456926809876641 330098558344280221506253224561
11112136131 108 123479569393875649161 1372121784502007371163027936091
11116321311 108 123572599489392758721 1373672721159604442219333403231
11133611121 108 123957296593654876641 1380092335884211352606120724561
11213161131 108 125734982549769199161 1409886619134035257053123011091
11311321611 108 127945996587475635321 1447238316240845405742382222131
11312161131 108 127964989453707199161 1447560579827051502203940011091
11613111117 126 134864349815788987689 1566194680132715995057302038613
21111316311 108 445687676383094648721 9409053512038115542200462588231
21113611311 108 445784582591987138721 9412122405283593350066231673231
26311111131 108 692274568947832099161 18214513116751532102554082861091
31111119141 108 967901734205496577881 30112506169647719011523576320221
31111121163 108 967901860018866472569 30112512040940020294012574877747
31111613211 108 967932476590869730521 30113940826060450950057153512931
31111711116 126 967938568565437965456 30114225123242464523285859208896
31114141113 144 968089777199676878769 30121281937843476384022059729897
31141141113 144 969770669819778878769 30199765276106264342202348729897
61112131311 108 3734692593372906578721 228235024172424195228533510433231
71311161111 126 5085281698998998754321 362637342532137407597073138410631
111121142313 144 12347908268945998989969 1372113672021417832276299508458297
112119111117 126 12570695077666192987689 1409415158230780836621069204038613
112143211113 144 12576099798734886698769 1410324214707683209378187604219897
112191112311 108 12586845681579415760721 1412132217503301577476813213336231
119121111117 126 14189839113748660987689 1690309401801207047924464338038613
121131211911 108 14672770499027588271921 1777330472639180014401320942051031
129111112131 108 16669679275703655361161 2152240830153181511261437173344091
132291111111 108 17500938078982947654321 2315218543953464022012076230260631
134111221113 144 17985819628419976958769 2412100233085567028743834643289897
141211112133 144 19940578189838699809689 2815831222762166800008919138856637
161361111111 108 26037408178976487654321 4201425114210285198645410640260631
161613111111 108 26118797682976431654321 4221140152024599402144077316260631
211111312911 108 44567986439006155293921 9408806130938233066110015407114031
211132113114 144 44576769187982890776996 9411587474453873625025426821125544
213211111134 144 45458977910994898765956 9692359191419184521295324171754104
221111113911 108 48890124694963217715921 10810149930551006160340101469277031
221111191131 108 48890158843369613059161 10810161256441248405201661481501091
221131111413 144 48898968434748618856569 10813083236925167057019832205921997
221131131114 144 48898977147757058880996 10813086127001155938432544598909544
221311111911 108 48978608255283166071921 10839510252830001223537306705751031
223111411113 144 49778701768834099898769 11106196395017765153207562041619897
231211141113 144 53458591774775598878769 12360222006539902106268143838729897
292131111111 108 85340586078947427654321 24930640234106850806012082150260631
311111133141 108 96790137164277028525881 30112489250051042810723218755322221
311111213214 144 96790186987486968209796 30112512500886986516764283239444344
311111314131 108 96790249780317760285161 30112541804222392461075080008910091
311113141122 144 96791386578797487418884 30113072312083479476410277419747848
311113411122 144 96791554579966593298884 30113150713174648766021468115787848
311114112312 144 96791990879683749985344 30113354321444009871981641569955328
311121241311 144 96796426794897492998721 30115424458897849222764009255363231
311212114113 144 96852979970682933776769 30141820654820280581077456096440897
311212141311 144 96852996899377832798721 30141828557443018948936490472063231
311221131114 144 96858592451877578880996 30144440700983282712399333518909544
311221141131 144 96858598686881819959161 30144443611680938167828178237351091
312211111911 108 97475778400702966071921 30433021158873710341350163405751031
312211411113 144 97475965229170699898769 30433108653801106646148515741619897
312341111112 144 97556969690678729876544 30471052309906301439064663546556928
314111112231 144 98665790826995877797361 30992021295818872475381222146622391
321121111911 108 103118768514956986071921 33113613604416005601752611735751031
332111112114 144 110297790789597877548996 36631121962850657295651054084137544
431111113212 144 185856791934889880956944 80124928469061440001982287681544128
431111121132 144 185856798763689976961424 80124932885018897501110751355211968
Sunday, January 19
Spotted
I don't take a lot of photographs of people I don't know, but when I do, I prefer to do it on the sly. So it was somewhat unfortunate that Weston's "pigeon man" decided to break the fourth wall and acknowledge me across the intersection. The upraised hand, possibly a gesture of "peace"; the piercing stare, not so much.
Saturday, January 18
Anagrammatic sums
Éric Angelini asked about anagrammatic sums on MathFun on January 12: "Let a + b = c and a < b < c and a, b, c = anagrams of each other." Halfway down his sausage article, he lists the Gilles Esposito-Farèse calculation for 3- to 5-digit results: 1 @ 3-digit, 25 @ 4-digit, and 648 @ 5-digit. In that spirit, here are 17338 @ 6-digit results. I count 495014 @ 7-digit and 17565942 @ 8-digit.
The idea for these has been around a few years. Claudio Meller's A160851 appears to be a (currently) somewhat misguided attempt at enumeration, while Rajesh Bhowmick's A203024, fleshed out by Charles Greathouse, provides a seemingly complete listing of sums, including 9449 6-digit terms. My 17338 6-digit results yield only 9443 distinct sums. Why six fewer?
Apparently 6-digit sums are the first that allow the sums to be twice one of the addends (i.e., a = b). In A023086 we see that there are twelve such. It turns out that six of these are the six that are not in my 17338 sums (because I did not allow a = b):
251748 = 2 * 125874
257148 = 2 * 128574
285174 = 2 * 142587
285714 = 2 * 142857
517482 = 2 * 258741
825174 = 2 * 412587
The idea for these has been around a few years. Claudio Meller's A160851 appears to be a (currently) somewhat misguided attempt at enumeration, while Rajesh Bhowmick's A203024, fleshed out by Charles Greathouse, provides a seemingly complete listing of sums, including 9449 6-digit terms. My 17338 6-digit results yield only 9443 distinct sums. Why six fewer?
Apparently 6-digit sums are the first that allow the sums to be twice one of the addends (i.e., a = b). In A023086 we see that there are twelve such. It turns out that six of these are the six that are not in my 17338 sums (because I did not allow a = b):
251748 = 2 * 125874
257148 = 2 * 128574
285174 = 2 * 142587
285714 = 2 * 142857
517482 = 2 * 258741
825174 = 2 * 412587
The other six are included because they each had an alternate solution:
517428 = 2 * 258714 = 241587 + 275841
571428 = 2 * 285714 = 142857 + 428571
571482 = 2 * 285741 = 158724 + 412758
825714 = 2 * 412857 = 241587 + 584127
851742 = 2 * 425871 = 127584 + 724158
857142 = 2 * 428571 = 142857 + 714285 = 275418 + 581724 = 285714 + 571428
Addendum: Éric and Gilles had created A331468 for their triples.
Addendum: Éric and Gilles had created A331468 for their triples.
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