Tuesday, February 20, 2024

Almost

Catherine had taken Bodie to Jameson Queen Animal Hospital in a taxi. I was following her progress on the Find-My app while watching Johnny Strides walk down Roncesvalles Ave on live TV, knowing that he wasn't far away from her and heading haphazardly in her direction.

She had already gone into the taxi for her return trip home when Johnny was still at Sorauren Ave. I hadn't realized it on the live stream but checking the You-Tube video, he managed to capture the taxi driving by! Catherine and Bodie are in the back seat:

click to enlarge

Monday, February 12, 2024

Confined (a loop)

I found a loop in Éric Angelini's "confined" sequence (about which I wrote last month). Term #60614674264 (= 27651356989742597468495745) is a duplicate of term #18563532230. Differences in the lead-up terms are highlighted here:

#18563532226   6912789247435649367123936   #60614674260   6912789247185649367123936
#18563532227  13825578494871298734247872   #60614674261  13825578494371298734247872
#18563532228  13825678494871298734247872   #60614674262  13825678494371298734247872
#18563532229  27651356989742597468495744   #60614674263  27651356988742597468495744
#18563532230  27651356989742597468495745 = #60614674264  27651356989742597468495745

So we have a loop of length 42051142034. The smallest term in the loop appears to be 507434154592, so here is an abridged loop sequence (asterisk denotes the largest term; three twelve-digit local minima are also shown; indices of all these corrected February 29):

          0                                                       507434154592
          1                                                      1014868309184
          2                                                      2029736618368
          3                                                      2029736718368
          4                                                      4059473436736
          5                                                      8118946873472
          6                                                      8128946873472
          7                                                     16257893746944
          8                                                     16257893746945
          9                                                     32515787493890
         10                                                     65031574987780
         11                                                     65031574987880
         12                                                     65031574987890
         13                                                    130063149975780
         14                                                    131631410075780
         15                                                     13163141175780
         16                                                     13163141275780
         17                                                     26326282551560
         18                                                     26326282561560
         19                                                     52652565123120
         20                                                    105305130246240
        ...                                                                ...
17074586421  49512395802029907136051366345193519491458782692496790312698501120
17074586422 495123958020210007136051367345193519491458782692496790312698501220 *
17074586423   4951239580202117136051367345193519491458782692496790312698501230
        ...                                                                ...
25756695203                                                      5007793970328
25756695204                                                       517893970328
25756695205                                                      1035787940656
        ...                                                                ...
25757984145                                                      5097006463136
25757984146                                                       509716463136
25757984147                                                      1019432926272
        ...                                                                ...
27813217917                                                      6806950060736
27813217918                                                       680695160736
27813217919                                                      1361390321472
        ...                                                                ...
42051142014                                                   1128050902650182
42051142015                                                   1228050902650182
42051142016                                                   1238050902650182
42051142017                                                   2476101805300364
42051142018                                                    247610180531364
42051142019                                                    495220361062728
42051142020                                                    495230361062728
42051142021                                                    990460722125456
42051142022                                                   1000460723125456
42051142023                                                     11460723125456
42051142024                                                     12460723125456
42051142025                                                     24921446250912
42051142026                                                     24921456250912
42051142027                                                     49842912501824
42051142028                                                     99685825003648
42051142029                                                     10068582513648
42051142030                                                      1168582513648
42051142031                                                      1268582513648
42051142032                                                      2537165027296
42051142033                                                      5074330054592
42051142034                                                       507434154592

Wednesday, February 07, 2024

Back from the vet


Ten-year-old Bodie is back from the veterinarian where, this morning, he had surgery to remove five teeth and a papilloma on his back.

Friday, January 26, 2024

A million-digit Leyland prime (end of 2nd run)

My second run of 59754 Leyland-prime candidates was started on 27 April 2023 and resulted in my finding three PRPs (the first column is the number of decimal digits):

1000905  (197180,119151)  Aug 2023
1000910  (191319,170462)  Jul 2023
1000999  (194968,136197)  Dec 2023

I blogged my first run here. As I did last year, I have documented my primality-test output. Search for PRP therein to situate the three primes. I have improved my average evaluation time per test from 9.45 hours to 5.55 hours but I was wrong in predicting that "I might be able to shave a couple of months off the total time required". It still took nine months. Some of that expected time saving was eaten up by the additional 218 tests. Some more, perhaps, by down-time as I have had a few battery-back-up units shutting themselves off, presumably because they are getting old. The remainder of the unmaterialized time saving would be an early start on a third search which began on some of my machines back on December 11.

Tuesday, January 09, 2024

Confined

In Éric Angelini's latest effort, he posits some interesting sequences. Specifically, half-way down the page, we have "replace the chunk by [the chunk + 1]". In case this is not entirely clear, allow me to restate the rule. Any integer that contains one or more blocks of identical adjacent digits evolves into another integer where each of these blocks is replaced with the value of the block plus one. Thus 133555777799999000000 becomes 13455677781000001. The two 3s are replaced with 34, the three 5s with 556, the four 7s with 7778, the five 9s with 100000, and the six 0s with 1. If our integer does not contain any blocks of identical adjacent digits, it becomes twice that integer. A starting integer evolves by the repeated application of these rules:

133555777799999000000
    13455677781000001
        1345667788111
        1345677889112
        1345678899122
       13456789100123
        1345678911123
        1345678911223
        1345678912233
        1345678912334
        1345678912344
        1345678912345
        2691357824690
        5382715649380
       10765431298760
       21530862597520
       43061725195040
       86123450390080
        8612345039180
       17224690078360
        1723469178360
...

Starting with the integer 1, Giorgos Kalogeropoulos makes the evolution out to be:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 65636, 131272, 262544, 262545, 525090, 1050180, 2100360, 211360, 212360, 424720, 849440, 849450, 1698900, 169891, 339782, 349782, 699564, 6100564, 611564, 612564, 1225128, 1235128, 2470256, 4940512, 9881024, 9891024, 19782048, 39564096, 79128192, 158256384, 316512768, 633025536, 634025636, 1268051272, 2536102544, 2536102545, 5072205090, 5072305090, 10144610180, 10145610180, 20291220360, 20291230360, 40582460720, 81164921440, 81264921450, 162529842900, 16252984291, 32505968582, ...

He suggests that the sequence seems to "explode to infinity". Actually, that initial explosion levels off after a thousand or so terms:

click to enlarge

I was sufficiently interested in this sequence to generate 15 billion terms. I graphed only the local minima and maxima, one each for every million terms. The initial explosion terms are ignored by setting the first minimum to 114782627657382. This way we see the sequence's confined space. Three extrema (one maximum, two 11-digit minima) are identified:

click to enlarge

The minima and maxima medians are ~10^18 and ~10^54. Because of the confined space the sequence will evolve into a loop, but particulars about this loop might never be known. To get a sense of this, be aware that in the graphed 15 billion sequence terms there are only 15 confined 11-digit integers. An additional 23 exist at the start but I cannot include these as being confined. So the sequence generates about one 11-digit integer every 10^9 terms. It could of course be more, or less, because the statistical estimate is based empirically on the 15 billion terms that we have so far examined.

How many random 11-digit integers are required in order to have a 50% chance that two of them are duplicates? It is roughly 350000. So, we need to generate some 350*10^3*10^9 = 350 trillion terms in order to have a decent shot at finding a loop.

Saturday, December 30, 2023

First digits after the decimal point

Éric Angelini is not at the moment updating his latest sequence suggestion, so I will post my extension (assuming a 1, 2 start) here:

1, 2, 11, 5, 42, 4, 94, 7, 74, 20, 27, 129, 101, 777, 7618, 1124, 14753, 1218, 82554, 8156, 98795, 3206, 32451, 499, 15377, 2366, 15386, 1868, 121402, 2419, 199254, 5819, 292038, 9247, 316636, 13812, 43621, 38327, 36725, 95818, 260900, 20134, 771711, 58457, 54269, 92835, 60177, 55065, 91504, 25771, 281635, 53566, 190196, 355062, 1866812, 199551, 10689395, 186879, 17482653, 95960, 54888692, 216986, 39532004, 138119, 34938527, 829143, 23731481, 189726, 7994697, 1908167, 23867906, 7480187, 31339937, 11404557, 36389852, 32664599, 23949435, 5535335, 23112591, 6504115, 6357184, 3851878, 6059094, 489221, 807416, 1650409, 2044061, 12385171, 10283218, 83028464, 807417055, 82135674, 101726454, 26620507, 261687159, 61339564, 2344003581, 48771083, 2080674423, 22994000, ...

click to enlarge

Thursday, December 21, 2023

Wednesday, December 13, 2023

A million-digit Leyland prime (third one)

After my second find, I really did not think that I would generate another example in my current search space. However, at the upper end of that space:

194968^136197+1*136197^194968 is 3-PRP!

This is now the 14th largest-known Leyland prime. Ryan Propper owns the largest 13:

19 million-digit Leyland primes (via Norman Luhn); n-th Leyland + 1 = A076980 index

Bucket list #1

My original five-piece bucket list is here with #2 here.

1. Wiener schnitzel (The Coffee Mill closed in 2014; Amber closed in 2021.)

Café Polonez: 195 Roncesvalles Ave
Catherine and I had lunch today at Café Polonez with Shelley and Denny Mylko. The restaurant is just north of Fern Ave. I of course had the pork schnitzel:

schnitzel with cole slaw, beets, carrots, and potatoes: $21.95 plus tax
Very nice!