Saturday, December 31, 2022
Saturday, December 24, 2022
Friday, December 23, 2022
Monday, December 19, 2022
Binary complement sequences
On Friday, Joshua Searle posted to the Sequence Fanatics Discussion list a neat procedure: take the binary complement of an integer multiplied by 3. Iterate. For example, starting with 3 we get the binary of 9 (1001), the complement of which (0110) is 6. Continuing, from 6 we get the binary of 18 (10010), the complement of which (01101) is 13. Arriving at zero, we stop.
0 3
1 6
2 13
3 24
4 55
5 90
6 241
7 300
8 123
9 142
10 85
11 0
Eleven steps to get to zero. The largest integer reached is 300 at step 7. We can shorthand the sequence data for 3 with (11,7,300) [steps to reach zero, steps to reach a maximum, the maximum]. Here are the statistics for integer starts up to 28:
0 (0,0,0)
1 (1,0,1)
2 (2,0,2)
3 (11,7,300)
4 (12,8,300)
5 (1,0,5)
6 (10,6,300)
7 (3,1,10)
8 (4,2,10)
9 (13,9,300)
10 (2,0,10)
11 (19,15,300)
12 (80,28,328536)
13 (9,5,300)
14 (2,1,21)
15 (15,11,300)
16 (16,12,300)
17 (81,29,328536)
18 (14,10,300)
19 (11,7,300)
20 (12,8,300)
21 (1,0,21)
22 (6,2,72)
23 (83,31,328536)
24 (8,4,300)
25 (73,21,328536)
26 (22,5,661)
27 (79,27,328536)
28 (7572,2962,123130640068522377168864228132316865867184046004226894)
Saturday, December 10, 2022
Mobile upgrade
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| iPhone 6 on warp drive |
Catherine's almost-seven-year-old iPhone 6 was dying/dead so I ordered a replacement.
Starting it up went well enough although I did spend some fifteen minutes prior looking for a paper clip. Only later did I discover the iPhone 13's accompanying SIM-tray ejector tool:
Thursday, December 08, 2022
Products with embedded indices
Éric Angelini did a "smallest multiplication" bit yesterday that I felt was worth extending.
0 0 = 0 * 1
1 10 = 2 * 5
2 12 = 3 * 4
3 132 = 6 * 22
4 84 = 7 * 12
5 152 = 8 * 19
6 126 = 9 * 14
7 170 = 10 * 17
8 198 = 11 * 18
9 195 = 13 * 15
10 1008 = 16 * 63
11 1100 = 20 * 55
12 1218 = 21 * 58
13 713 = 23 * 31
14 1416 = 24 * 59
15 1150 = 25 * 46
16 1612 = 26 * 62
17 1728 = 27 * 64
18 1820 = 28 * 65
19 1914 = 29 * 66
20 1020 = 30 * 34
21 1216 = 32 * 38
22 1221 = 33 * 37
23 2345 = 35 * 67
24 2448 = 36 * 68
25 2925 = 39 * 75
26 12600 = 40 * 315
27 1927 = 41 * 47
28 2898 = 42 * 69
...
The column of indices on the far left is shown embedded (in bold) in their adjacent products. The constraint on the multiplier and multiplicand is that they must be distinct nonnegative integers with the multiplier the smallest such not yet used and the multiplicand the smallest such that yields the embedded index. A chart extending the products is here.
Friday, November 18, 2022
Coyote sighting
Spotted early this morning walking west down our street, just as Bodie and I were leaving for our walk. Already too distant to be distinct by the time I managed the photo, I have added a helpful arrow. At the bottom of the shot are some of the two dozen bags of leaves where our driveway meets the road, ready for today's yard-waste pickup. Resourceful squirrels will tear the paper off the bottom of the bags in order to steal the mostly dry leaves. I've had to re-bag one of them because it was beyond repair.
Wednesday, November 16, 2022
Paper delivery
Saturday, November 05, 2022
Friday, November 04, 2022
Twin concatenations
Éric Angelini came up with this idea of listing products (one per line) where the multipliers introducing the line are the positive integers and the smallest possible multiplicands (after the initial line) are deduced from the products, given that the concatenations of the products (to the right of the equal signs) must be the same as the concatenations of the multipliers and multiplicands (in order, to the left of the equal signs), from the beginning, as far as it will go.
On the MathFun mail list, Éric shared Gilles Esposito-Farese's version that makes the multipliers the nonnegative integers (up to 50), instead of the positive-integers versions shown on Éric's blog:
0 * 0 = 0
1 * 0 = 0
2 * 5 = 10
4 * 63659 = 254636
5 * 119 = 595
...
For some reason Gilles skipped 3. Which makes his output for lines 4 to 50 incorrect. Ouch!
Here is my correction:
0 * 0 = 0
1 * 0 = 0
2 * 5 = 10
3 * 846 = 2538
4 * 1 = 4
5 * 1283 = 6415
6 * 2 = 12
7 * 1194673 = 8362711
8 * 118342264792783 = 946738118342264
9 * 88 = 792
10 * 7839881 = 78398810
11 * 7127164651933786 = 78398811171271646
12 * 432781551 = 5193378612
13 * 332908885487176222196 = 4327815511333290888548
14 * 512587299724651848 = 7176222196145125872
15 * 664831 = 9972465
16 * 11550979 = 184815664
...
Friday, October 14, 2022
Non sequitur
I don't watch a lot of Netflix, likely because after wading through a bunch of popular/trending items I don't find anything palatable. Yesterday I tried a different approach. I looked through the critically acclaimed films and (eventually) came across The House:
Although the movie left me slightly discombobulated, I thought it generally worthwhile. For some reason, about half-way through, Harry Mathews' The Sinking of the Odradek Stadium came to mind. I once owned a copy of it. In the early 1980s (I think), I lent it to Brad (I think), a workplace acquaintance. I lost track of Brad at some point. I never got the book back!
Tuesday, September 13, 2022
Delicious Dishes (one-reeler)
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.jpg) |
| 27 March 1943: click to enlarge |
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| 5 July 1947: click to enlarge |
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| 5 November 1950: click to enlarge |
.jpg) |
| N.K. Morris Mfg. Co. advert in The Billboard, 30 June 1951 |
.jpg) |
| N.K. Morris Mfg. Co. advert in The Billboard, 7 April 1956 |
Sunday, September 11, 2022
A million-digit Leyland prime (spot check)
As expected, one of my slower processes has rediscovered (this afternoon) Gabor Levai's 1000027-digit Leyland prime. I thought that it might be a good time to see how far I've come. It seems that I've completed about 49% of the numbers that I wanted to test since I started 132 days ago, so, on average, I'm testing about 220 numbers per day. It means that, at best, I'll be done by February 2023 but a lot depends on my transferring processes from my three slow computers to others as they become available.
I had installed differing operating systems on those machines to see what would happen. I saw no improvement. So I'm thinking now that the slow speed arises from the fact that these three Mac minis were purchased (in order to save money) with only 8 GB RAM, which may be insufficient for the task at hand (it had been adequate for testing 300000-digit numbers). I will now terminate three of the six processes on each machine to see if there is any speed improvement on the remaining three processes. Later this month I will have completed 14 processes on two other machines, which allows me to transfer the 9 terminated processes.
Thursday, September 01, 2022
A million-digit Leyland prime (power interruption)
This morning, a 23-minute power interruption allowed a couple of Toronto Hydro employees to do some maintenance work on a close-to-my-home transformer. I have all 18 computers (104 processes currently working on my million-digit Leyland prime search) on battery back-up units but the amount of time here was just long enough to shut down seven of them (39 processes). Another 5 to 10 minutes might have killed them all.
By early afternoon I had saved the output of the interrupted programs and reinitialized their new search ranges. Three of the seven affected computers were the ones that were taking 18.5 hours per candidate in my May 27 reality check, so I installed new (differing) operating systems on those machines, hoping for a speed improvement. I won't know for a couple of days if that will pan out.
Saturday, August 27, 2022
The tooth abscess
Back on April 20, I developed a terrible toothache that came to be associated with a swollen lower-right jaw. Over the next five days I consumed more Acetaminophen and Ibuprofen pills than likely I had over several decades. On April 25, my dentist (whom I had not seen since December 2019 because of the pandemic) had a look.
He prescribed a seven-day regimen of Amoxicillin. Within 24 hours the situation improved. He also recommended that I see an endodontist, which I did on May 10. The endodontist gave me three options: do nothing, have a certain dead tooth pulled, or let him do a root canal that involved (at least, pending complications) two three-hour sessions. I don't know that I can last one three-hour session. The last time I had a root canal the doctor left me alone for ten or so minutes and I chose that time to have a panic attack! Unfortunately, that set in motion similar feelings of anxiety, subsequently, every time that I have had my mouth worked on in a dentist's chair.
Mulling over my options, I did nothing over the course of the summer. On July 7, I managed to complicate things by breaking off part of a tooth in the upper-left part of my mouth. As annoying as this was, I did not see my dentist about it. Around August 20, I noted that the area around my dead tooth was getting reinfected. Salt-water rinses were not helping. Fortunately, this time the formation of the abscess was not accompanied by pain and, by August 22, I felt that my own body was dealing with it and that, perhaps, the worst was over. Nevertheless, I was fearful that my immune system was not capable of conquering the bacterial pathogen and I thought it prudent to see if I could acquire an antibiotic.
Constrained somewhat by my unwillingness to travel any distance, I thought that making an appointment with my family doctor right here in Weston was an option. He could write me a prescription and I could get it filled nearby. But my call to his office on August 23 did not go as planned. The receptionist, upon discovering that my last appointment had been way back in November 2012 (I was as surprised as she was: Had it really been that long?) felt that she needed to ask the doctor if he wanted to engage with me. The next day she left a message with their (apparently new) policy: Due to being overburdened with Covid, my family doctor was not taking back patients he hasn't seen in over five year!
On August 25, I decided to give my dentist a call. My wife had suggested that they might be able to have a repeat of my April 25 prescription filled at a local dispensary and I should ask about that. They were busy and I had to leave a message. I gave my name and mentioned my previous appointment with the dentist on April 25 and asked about the possibility of filling a repeat of the prescription that I got then, but locally. I forgot to leave my phone number. There was no reply, so the next morning, Friday, I phoned again. The assistant/receptionist who answered tied my request to the previous day's call by noting that her inability to parse my full name had her calling back someone else for the reply. I provided the local Shoppers Drug Mart address and phone number. She said that she still had contact the dentist about all this but if she didn't get back to me that day, she would get back to me on Monday.
I spent the rest of Friday making my potato/carrot/spinach soup complemented with some lentils, roast-beef gravy, and leftover frozen bits from a long-ago spiral ham. I had several bowlfuls and tried to get some late-afternoon shut-eye. When I came to later that evening, I was feeling much better! The swelling in my gums had receded a little and I had the sense that my immune system was going to be able to handle this after all. Perhaps it was some remnant antibiotic in the ham!
Update: Monday came and went without hearing again from the receptionist. It appears that I have lost my dentist!
Saturday, August 20, 2022
Bucket list #4
My original five-piece bucket list is here.
4. Tim Hortons (I know. How can something so ubiquitous be so difficult to reach.)
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| 1931 Weston Road, this morning (after visiting Shoppers Drug Mart) |
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| Six doughnuts: $6.49 |
Saturday, August 13, 2022
The new backyard residents
The area of our backyard shown in the above has become a focus of activity of late, first by Bodie wading into the growth sniffing out something-or-other, then by the scurrying of some smallish animals (that I took to be mice) into the crevasses of the railway tie that is there (on the right), as well as the corners of the shed (on the far left) which sported some previously undisclosed openings into its underpinnings.
It wasn't just mice. Unperturbed by my presence, this afternoon I spotted a somewhat larger intruder, lazily munching away at some item in the (left) grassy area:
Tuesday, August 02, 2022
Wikipedia (reprise)
It has been more than a decade since I last made a Wikipedia edit. Just now I tried again. I changed Michelle Josef's birth year from 1954 to 1953. I have a December 1998 Globe & Mail article describing Josef as "a 45-year-old". And if that isn't substantive enough, I have a 15 March 2012 tweet from Michelle Josef herself saying: "I have been alive 21549 days."
Assuming that Michelle made the tweet on her 59th birthday (i.e., birthday anniversary), this would have her being born on 15 March 1953. How did Michelle arrive at the day count? 59*365 = 21535. Add 15 days for the leap years 1956 to 2012 and we have 21550 days. Why did Michelle calculate one day less? Perhaps she knew that century years (ending in 00) were not leap years and therefore did not count the year 2000 as one. Unfortunately, century years divisible by 400 are exceptions to the century-years rule and 2000 was a leap year. There are of course other possibilities. For example, (2012-1956)/4 = 14, or Michelle tweeted the day before her birthday, or some other scenario.
Update: On 18 October 2022, a Rickyharder added "March 21" to my "1953" Wikipedia year correction. My own investigation suggested that the date was March 20, based on a family source. I don't know which date is correct. Assuming my date, Michelle tweeted five days before her birthday, so her day count was four days too many. Go figure.
Monday, August 01, 2022
Sum & erase (integer version)
One of the drawbacks of Éric Angelini's sum-and-erase is that we are dealing with number strings, not integers. This is because we allow leading zeros. There's an easy and obvious fix: Don't allow leading zeros! Everything else is the same but, if the erasure of all instances of a digit (1-9) that is contained in the sum-of-digits moves one or more zeros to the leading (left) edge then those zeros are erased as well. Now the only number that is allowed to start with zero is zero itself. And 0 by itself disappears because it is the left-most digit of 0 and the sum-of-digits is 0. And although total erasure (the empty string "" in Éric's version) happens here as well, we can define it to be zero so as to keep our sequencing in the integer realm. Conveniently in Mathematica, FromDigits[{}] == FromDigits[{0}] == 0.
0 => "" = 0
1 => "" = 0
2 => "" = 0
3 => "" = 0
4 => "" = 0
5 => "" = 0
6 => "" = 0
7 => "" = 0
8 => "" = 0
9 => "" = 0
100..00 => 0 => "" = 0
Summarizing the procedure:
0. Start with a base-ten nonnegative integer.
1. Total the number's digits and concatenate this sum to the right of the number.
2. If the first digit in our number is one of the digits in our sum, delete every instance of it.
3. If (after deletion) there now appear leading zeros in our number, delete these as well.
4. If all digits or even all non-zero digits have been deleted, the subsequent term is zero.
5. Otherwise, the subsequent term is the concatenation of the remaining digits.
6. To generate a sequence of terms, iterate. Most starts eventually end up at zero.
In Éric's procedure, I had found (in addition to the fixed-point c0) six cycles. In this integer version, I have [index. name = c&start-string (cycle length) {smallest precursor}]:
0. c0 (1) {0}
1. c86 (80858) {16}
2. c30323322046333587 (1634) {5916}
3. c48822444886224973 (29) {23675}:
0 48822444886224973
1 4882244488622497385
2 488224448862249738598
3 488224448862249738598115
4 488224448862249738598115122
5 488224448862249738598115122127
6 488224448862249738598115122127137
7 882288622973859811512212713718
8 882288622973859811512212713718137
9 2262297359115122127137113714
10 226229735911512212713711371497
11 226229735911512212713711371497113
12 226229735911512212713711371497113118
13 6973591151171371137149711311818
14 6973591151171371137149711311818123
15 6973591151171371137149711311818123129
16 6973591151171371137149711311818123129141
17 6973591151171371137149711311818123129141147
18 6973591151171371137149711311818123129141147159
19 6973591151171371137149711311818123129141147159174
20 97359115117137113714971131181812312914114715917418
21 73511511713711371471131181812312141147151741818
22 73511511713711371471131181812312141147151741818153
23 73511511713711371471131181812312141147151741818153162
24 351151113113141131181812312141141514181815316211
25 351151113113141131181812312141141514181815316211124
26 51151111114111181812121411415141818151621112411
27 111111114111181812121411411418181162111241111
28 11111111411118181212141141141818116211124111197
29 48822444886224973
4. c77078088077807837313303137333430853003033013 (43) {83123}
5. c886054637 (140) {256311}
6. c5869099118496111114109711 (85) {346715}
Also six cycles (so far), in order of smallest precursor. I have a distance-to-cycle chart. For positive terms it is the number of steps needed to get to zero. Negative terms are for starts that will reach some other cycle. Terms that are 0 are for integers that are part of a cycle.
Sunday, July 31, 2022
Bodie's weight
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| click to enlarge |
The vertical scale is kilograms. Red dots are vet readings. Blue dots are home (bathroom scale) readings. I don't really trust the bathroom scale to be accurate but it will do. We just want to be cognizant of inordinate weight gain. So every two weeks Bodie looks forward to jumping into my arms. Unfortunately, the process also monitors my personal weight gain.
Saturday, July 30, 2022
Aerial action
Just beyond my neighbour's yard, a northern mockingbird lets a hawk know it's not welcome.
Friday, July 29, 2022
Don't it always seem to go ...?
It has been a year since I last received delivery of a couple of big bottles of Welch's grape juice. I should have ordered more in the following months! By the time I did, they were out of stock. I can't tell if the supply shortage is temporary or permanent.
Thursday, July 28, 2022
Cycles in Éric Angelini's sum-and-erase
Éric's article on sum-and-erase is here. I wanted a place where I could post known cycles, so I will do that in this listing [index. name = c&start-string (cycle length) {precursors}]:
0. c0 (1) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 00, 01, 02, 03, 05, 06, 07, 08, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, ...}
1. c3374 (583792) {04, 09, 25, 31, 63, 67, 69, 77, 92, 96, 99, 023, 027, 031, 040, 042, 044, 049, 067, 074, 075, 081, 082, 090, 093, 099, 105, 109, 116, 133, 138, 148, 152, 161, 162, 163, 168, 174, 194, 197, 198, 206, 216, 253, 257, 258, 285, 307, 314, 327, 352, 370, 382, 389, 398, 422, 423, 430, 463, 477, 482, 483, 496, 503, 521, 543, 544, 568, 581, 582, 590, 592, 599, 604, 605, 610, 613, 630, 635, 639, 645, 656, 665, 667, 668, 676, 681, 682, 685, 686, 688, 708, 722, 736, 737, 742, 758, 766, 773, 782, 801, 809, 810, 814, 821, 839, 841, 848, 849, 856, 858, 867, 875, 884, 885, 894, 902, 906, 923, 932, 944, 958, 977, 982, 988, 0119, 0135, 0144, 0147, 0152, 0163, 0166, 0176, 0191, 0199, 0203, 0207, 0216, 0230, 0235, 0244, 0253, 0256, 0259, 0267, 0270, 0275, 0279, 0299, 0301, 0310, 0314, 0317, 0333, 0349, 0361, 0375, 0378, 0385, 0387, 0400, 0402, 0404, 0409, 0420, 0426, 0429, 0440, 0448, 0482, 0484, 0486, 0490, 0498, 0521, 0539, 0559, 0562, 0578, 0592, 0595, 0607, 0619, 0645, 0656, 0658, 0665, 0668, 0670, 0671, 0673, 0686, 0698, 0704, 0705, 0713, 0716, 0735, 0736, 0738, 0740, 0750, 0767, 0768, 0776, 0794, 0801, 0802, 0810, 0813, 0814, 0817, 0819, 0820, 0824, 0829, 0831, 0847, 0855, 0856, 0866, 0867, 0900, 0903, 0909, 0911, 0916, 0919, 0925, 0929, 0930, 0938, 0957, 0958, 0982, 0990, 0991, 0992, 0999, 1005, 1009, 1016, 1019, 1027, 1050, 1056, 1059, 1061, 1062, 1082, 1090, 1091, 1099, 1106, 1109, 1119, 1120, 1163, 1168, 1191, 1192, 1201, 1210, 1222, 1238, 1239, 1250, 1253, 1254, 1264, 1268, 1277, 1285, 1289, 1299, 1307, 1327, 1333, 1335, 1337, 1339, 1352, 1353, 1364, 1365, 1370, 1373, 1376, 1378, 1393, 1396, 1422, 1495, 1520, 1526, 1528, 1530, 1545, 1554, 1557, 1559, 1566, 1575, 1578, 1593, 1595, 1604, 1613, 1618, 1628, 1629, 1630, 1631, 1637, 1648, 1649, 1668, 1679, 1681, 1685, 1686, 1724, 1728, 1729, 1739, 1754, 1760, 1762, 1767, 1776, 1792, 1793, 1794, 1823, 1825, 1832, 1840, 1843, 1853, 1866, 1867, 1894, 1904, 1907, 1908, 1911, 1912, 1921, 1923, 1929, 1933, 1934, 1938, 1970, 1987, 1992, 2006, 2019, 2028, 2047, 2048, 2053, 2058, 2060, 2068, 2074, 2082, 2085, 2091, 2094, 2099, 2109, 2126, 2133, 2134, 2157, 2162, 2163, 2168, 2169, 2173, 2188, 2194, 2208, 2216, 2222, 2230, 2234, 2249, 2254, 2264, 2280, 2291, 2295, 2299, 2302, 2307, 2314, 2320, 2324, 2342, 2344, 2347, 2348, 2351, 2357, 2358, 2361, 2370, 2384, 2389, 2391, 2394, 2395, 2398, 2413, 2414, 2416, 2418, 2429, 2441, 2458, 2471, 2480, 2481, 2492, 2503, 2524, 2531, 2542, 2545, 2554, 2556, 2565, 2576, 2584, 2604, 2624, 2642, 2643, 2644, 2655, 2668, 2679, 2686, 2687, 2704, 2740, 2757, 2758, 2763, 2764, 2775, 2777, 2784, 2786, 2793, 2802, 2805, 2818, 2820, 2837, 2839, 2843, 2848, 2850, 2856, 2875, 2881, 2884, 2897, 2904, 2906, 2912, 2917, 2918, 2921, 2925, 2929, 2948, 2951, 2952, 2964, 2992, 3007, 3019, 3033, 3054, 3057, 3061, 3070, 3071, 3089, 3091, 3095, 3098, 3109, 3117, 3125, 3129, 3136, 3140, 3146, 3148, 3163, 3164, 3167, 3171, 3176, 3177, 3199, 3213, 3218, 3231, 3237, 3244, 3254, 3256, 3273, 3287, 3298, 3299, 3303, 3316, 3321, 3327, 3330, 3335, 3352, 3353, 3371, 3374, 3387, 3396, 3398, 3457, 3462, 3476, 3502, 3509, 3512, 3515, 3523, 3528, 3532, 3533, 3546, 3551, 3564, 3578, 3604, 3611, 3612, 3619, 3644, 3654, 3655, 3668, 3675, 3686, 3687, 3700, 3701, 3707, 3709, 3713, 3724, 3731, 3734, 3743, 3747, 3757, 3767, 3770, 3774, 3775, 3776, 3789, 3792, 3794, 3804, 3809, 3812, 3814, 3821, 3837, 3851, 3855, 3856, 3858, 3859, 3866, 3872, 3873, 3885, 3888, 3896, 3905, 3911, 3922, 3929, 3936, 3938, 3942, 3950, 3963, 3967, 3969, 3972, 3983, 3986, 3989, 3992, 3996, 3998, 4019, 4023, 4035, 4053, 4063, 4069, 4091, 4096, 4097, 4099, 4102, 4109, 4123, 4139, 4146, 4155, 4163, 4164, 4176, 4196, 4202, 4218, 4220, 4228, 4235, 4239, 4253, 4282, 4283, 4299, 4300, 4301, 4307, 4347, 4348, 4349, 4361, 4368, 4370, 4371, 4374, 4376, 4378, 4380, 4381, 4384, 4387, 4394, 4399, 4416, 4437, 4438, 4439, 4452, 4458, 4461, 4483, 4491, 4492, 4506, 4516, 4524, 4527, 4531, 4542, 4548, 4578, 4584, 4598, 4612, 4614, 4616, 4625, 4627, 4628, 4641, 4661, 4668, 4671, 4686, 4698, 4701, 4710, 4731, 4767, 4776, 4793, 4796, 4815, 4826, 4834, 4843, 4850, 4866, 4888, 4890, 4911, 4914, 4922, 4924, 4929, 4935, 4938, 4939, 4941, 4942, 4959, 4960, 4970, 4992, 4993, 4995, 5003, 5019, 5030, 5031, 5033, 5034, 5035, 5038, 5039, 5053, 5063, 5067, 5088, 5091, 5109, 5113, 5131, 5134, 5142, 5163, 5166, 5176, 5186, 5196, 5198, 5218, 5236, 5244, 5249, 5263, 5264, 5274, 5286, 5296, 5299, 5307, 5333, 5339, 5361, 5366, 5370, 5371, 5384, 5388, 5391, 5393, 5428, 5429, 5439, 5446, 5448, 5455, 5456, 5464, 5465, 5483, 5484, 5487, 5493, 5496, 5503, 5545, 5546, 5554, 5555, 5562, 5566, 5578, 5582, 5584, 5604, 5616, 5625, 5630, 5637, 5648, 5652, 5656, 5661, 5665, 5668, 5686, 5694, 5720, 5742, 5758, 5760, 5761, 5763, 5767, 5776, 5779, 5781, 5782, 5785, 5794, 5797, 5810, 5814, 5825, 5827, 5832, 5845, 5852, 5854, 5863, 5866, 5870, 5871, 5900, 5929, 5930, 5937, 5938, 5964, 5979, 5982, 5992, 5997, 6004, 6005, 6014, 6019, 6021, 6040, 6041, 6050, 6056, 6065, 6078, 6091, 6099, 6100, 6104, 6106, 6107, 6109, 6130, 6134, 6155, 6160, 6174, 6177, 6225, 6235, 6236, 6237, 6244, 6252, 6253, 6263, 6268, 6277, 6286, 6287, 6288, 6298, 6299, 6300, 6307, 6309, 6318, 6321, 6351, 6370, 6372, 6377, 6379, 6404, 6405, 6416, 6417, 6429, 6430, 6440, 6453, 6461, 6473, 6485, 6489, 6490, 6499, 6545, 6554, 6559, 6569, 6572, 6578, 6588, 6595, 6596, 6605, 6610, 6623, 6628, 6641, 6659, 6680, 6682, 6691, 6698, 6705, 6713, 6714, 6718, 6723, 6743, 6753, 6781, 6794, 6803, 6804, 6806, 6812, 6823, 6826, 6830, 6834, 6837, 6852, 6854, 6860, 6862, 6873, 6894, 6899, 6904, 6913, 6915, 6916, 6922, 6923, 6929, 6938, 6942, 6949, 6961, 6968, 6975, 6986, 6992, 6994, 6999, 7008, 7019, 7035, 7037, 7056, 7067, 7073, 7076, 7078, 7080, 7084, 7087, 7091, 7099, 7102, 7109, 7147, 7150, 7158, 7163, 7174, 7178, 7187, 7195, 7205, 7227, 7230, 7235, 7240, 7244, 7253, 7260, 7263, 7272, 7279, 7288, 7294, 7297, 7299, 7305, 7318, 7340, 7358, 7361, 7365, 7366, 7377, 7385, 7458, 7490, 7493, 7495, 7499, 7502, 7508, 7516, 7517, 7526, 7529, 7530, 7542, 7547, 7561, 7563, 7571, 7574, 7578, 7581, 7587, 7604, 7611, 7614, 7620, 7621, 7626, 7627, 7635, 7655, 7657, 7662, 7668, 7672, 7675, 7680, 7682, 7686, 7703, 7706, 7708, 7714, 7718, 7722, 7729, 7737, 7751, 7754, 7758, 7762, 7765, 7773, 7798, 7824, 7825, 7833, 7841, 7849, 7860, 7863, 7866, 7892, 7904, 7909, 7912, 7920, 7929, 7931, 7936, 7938, 7940, 7941, 7942, 7945, 7950, 7958, 7978, 7987, 7990, 7992, 7999, 8001, 8009, 8010, 8012, 8013, 8017, 8019, 8024, 8029, 8031, 8041, 8047, 8053, 8067, 8090, 8091, 8100, 8108, 8109, 8129, 8143, 8152, 8156, 8160, 8162, 8163, 8173, 8179, 8180, 8204, 8224, 8229, 8234, 8235, 8239, 8242, 8244, 8253, 8256, 8279, 8292, 8299, 8307, 8313, 8323, 8324, 8331, 8332, 8341, 8344, 8358, 8361, 8369, 8370, 8374, 8385, 8391, 8401, 8402, 8403, 8407, 8408, 8422, 8436, 8446, 8452, 8456, 8464, 8480, 8492, 8506, 8517, 8518, 8527, 8531, 8534, 8537, 8561, 8562, 8566, 8581, 8604, 8626, 8633, 8644, 8647, 8657, 8662, 8669, 8672, 8679, 8692, 8696, 8707, 8716, 8723, 8724, 8726, 8727, 8731, 8736, 8742, 8765, 8767, 8770, 8772, 8776, 8788, 8791, 8794, 8810, 8835, 8840, 8851, 8878, 8887, 8891, 8893, 8905, 8918, 8929, 8938, 8944, 8952, 8961, 8976, 8981, 8983, 8992, 9002, 9006, 9017, 9020, 9025, 9029, 9038, 9044, 9058, 9060, 9077, 9088, 9092, 9104, 9105, 9128, 9143, 9144, 9149, 9163, 9178, 9180, 9182, 9189, 9194, 9198, 9211, 9227, 9229, 9235, 9244, 9247, 9253, 9272, 9284, 9288, 9292, 9302, 9307, 9336, 9347, 9361, 9362, 9363, 9370, 9377, 9379, 9382, 9389, 9397, 9398, 9407, 9416, 9435, 9444, 9462, 9468, 9475, 9478, 9483, 9508, 9516, 9518, 9556, 9565, 9567, 9568, 9578, 9583, 9604, 9615, 9617, 9618, 9626, 9633, 9641, 9645, 9658, 9662, 9668, 9686, 9688, 9689, 9698, 9704, 9707, 9709, 9710, 9749, 9750, 9761, 9767, 9770, 9776, 9790, 9794, 9801, 9866, 9870, 9902, 9914, 9918, 9922, 9937, 9938, 9968, 9970, 9974}
2. c083433 (120621) {092, 245, 351, 395, 413, 433, 468, 571, 802, 0382, 0479, 0588, 0594, 0691, 0848, 0884, 0902, 0920, 1038, 1142, 1170, 1369, 1380, 1412, 1421, 1433, 1479, 1482, 1560, 1584, 1645, 1666, 1701, 1710, 1802, 1848, 1884, 1986, 2096, 2148, 2235, 2251, 2325, 2352, 2450, 2477, 2512, 2521, 2555, 2601, 2634, 2649, 2688, 2761, 2955, 2970, 2999, 3358, 3479, 3510, 3519, 3538, 3544, 3583, 3699, 3702, 3769, 3779, 3797, 3802, 3848, 3884, 3917, 3949, 3981, 3994, 4012, 4103, 4112, 4121, 4122, 4138, 4160, 4165, 4183, 4286, 4650, 4707, 4770, 4790, 4792, 4802, 4825, 4870, 4909, 4933, 4990, 5078, 5108, 5277, 5427, 5479, 5602, 5786, 5802, 5808, 5823, 5848, 5880, 5884, 6335, 6353, 6385, 6411, 6456, 6465, 6479, 6491, 6591, 6645, 6669, 6694, 6696, 6802, 6848, 6884, 6946, 6964, 6966, 7014, 7098, 7228, 7282, 7291, 7444, 7579, 7588, 7597, 7759, 7802, 7816, 7848, 7884, 7903, 7946, 8002, 8020, 8021, 8288, 8376, 8428, 8479, 8482, 8708, 8780, 8828, 8842, 8870, 8882, 9134, 9168, 9273, 9277, 9515, 9551, 9581, 9802, 9806, 9848, 9884, 9999}
3. c222227222772202078 (1723) {695, 825, 905, 0655, 0857, 0864, 1385, 1857, 2695, 3078, 3285, 3857, 4189, 4857, 5778, 5787, 6054, 6504, 6685, 6856, 6857, 6865, 7864, 8265, 8459, 8572, 9005, 9050, 9857}
4. c332 (14072) {323, 332, 3238, 3283, 3328, 5133, 7132}
5. c3933 (10538) {333, 3339, 3393, 3933}
6. c5464644657500000011711019071641751 (49) {243, 2439, 6782}
Roughly, in order of precursor frequency. Strictly speaking, the bold numbers are not precursors (to cycles) but, rather, cycle terms themselves. I include them for the sake of completeness. Are there any more cycles?
Update: In a 14 Sep 2023 video seminar, Neil Sloane @38:33 mentions Michael Branicky finding (in my notation) c00020203143 (20173) and c00660657510634435617071915 (46), precursors not mentioned.
Saturday, July 23, 2022
My tired lip, Angela
I don't recall when exactly I read Martin Gardner's article on mnemonics [Scientific American, October 1957; 'The Scientific American Book of Mathematical Puzzles & Diversions', 1959; reprinted as 'Hexaflexagons and Other Mathematical Diversions', 1988] but it was likely sixty years ago when I lived on Claremont Street and came across the 1959 book on puzzles and diversions, borrowed from what must have been the 'Boys & Girls Library' at Manning Ave. & Robinson St. I didn't get into Martin Gardner and Scientific American as an obsession until a few years later, December 1967 being my first store-purchased issue.
I was reminded of this by Futility Closet's recent 'Words and Numbers' which adjusts slightly Gardner's version (above) and gets into some π mnemonics. I had myself created one of these back then: "My tired lip, Angela, my lip." Twelve digits. Not too shabby and I recalled it after all these years, although I did attempt at first to place the 'tired' in front of the second 'lip'. I also remembered that the final p=9 wasn't actually π's twelfth digit but a rounding-up approximation. There is a much more intensive explanation/application of this particular use of mnemonics, here.
Saturday, July 16, 2022
The fire at #3
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| 1:25 AM (click to enlarge) |
.jpg) |
| aftermath, in the light of day |
Friday, July 01, 2022
The Townsend fire
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| Gray at age 14 |
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| Gray at age 19 |
Tuesday, June 28, 2022
Less is more
Last month I described my disappointment at the real-time performance I was getting in my million-digit Leyland-prime search. I have now done some testing on my wife's recently-acquired iMac, a 3.7 GHz 10-core Intel i9, slightly faster than my own two 3.6 GHz versions. Her computer was running only 4 of the million-digit search processes, because it was a replacement for a failing older 4-core iMac. Supplementing those processes by additional ones, I wanted to see how that would impact search time:
4 processes 4.2 hours 1.05 hours/process
5 processes 4.9 hours 0.98 hours/process
6 processes 5.9 hours 0.98 hours/process
7 processes 7.5 hours 1.07 hours/process
8 processes 9.7 hours 1.21 hours/process
In other words, on a 10-core iMac I can run 8 searches significantly faster simply by doing 4 at a time! Running 6 processes appears to be ideal.
Monday, June 27, 2022
Animal sightings
It's been an incredible couple of weeks for me spotting by my home animals that are usually seen much further afield. On June 11, it was ravens in our backyard maple tree. And on the same day, a chipmunk made it momentarily onto the edge of our deck. Both are rare in our neighbourhood and, as far as I know, never on our street. Blue jays are generally in flocks flying overhead but I was peripherally aware of (more by sound than by sight) a mating pair in the area. The photo, taken on June 18, shows one in our mulberry tree that could be an offspring of that pair. On June 20, the neighbour's side-yard hummingbird feeder, just over the fence, sported a hummingbird. And on June 23, I saw an opossum [caveat] crossing the road to the edge of our driveway. I don't think I've ever encountered a hummingbird or an opossum. As fate would have it, only the blue jay managed to make it into my camera.