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.

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.

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.

10+9+8*7*6*5+4+321

Happy New Year!

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

Dangerous obsession

wearing clogs on the stove and the countertop

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!

Neheim-Hüsten street view

Three years ago I wrote an article about my Neheim birthhouse, captured from a distance on a bicycle-ride YouTube video. In it, I bemoaned the absence of Google's street view. Today, by chance, I noticed that Google has (from October 2022) a very limited street-view coverage of Neheim (and its 1950s-era twin town of Hüsten):

baby-blue roads have street view: click to enlarge

I was fortunate that my birthhouse was visible from Bergheimer Weg (in that 2009 blog post is a link to a 1934 photograph of the house being built):

Bergheimer Weg 16, October 2022

Swap

Éric Angelini's blog post last Friday (Underline, reproduce) motivated me to attempt to solve

S9 = ninth, twentieth, thirtieth, fourth, ninetieth, sixth, seventh, eighth, ...

... wondering how many swaps (of "first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh, …") were needed to produce it. It stymied me for a day or so but (using a combination of manual and Mathematica step-throughs) I eventually got this:

S9 = ninth, twentieth, thirtieth, fourth, ninetieth; sixth, seventh, eighth, first, tenth, second, twelfth, fifteenth, thirtyeighth; thirteenth, fifth, eleventh, eighteenth, nineteenth, seventeenth, twentyfirst, twentysecond, fourteenth; twentyfourth, twentyfifth, twentysixth, twentyseventh, twentyeighth, sixteenth; third, thirtyfirst, thirtysecond, thirtythird, thirtyfourth, thirtyfifth, thirtysixth, thirtyseventh, twentythird; thirtyninth, fortieth, fortyfirst, fortysecond, twentyninth; fortyfourth, fortyfifth, fortysixth, fortyseventh, fortyeighth, fortyninth, fortythird; fiftyfirst, fiftysecond, fiftythird, fiftieth, fiftyfifth, fiftyfourth; fiftyseventh, fiftyeighth, fiftyninth, sixtieth, sixtyfirst; sixtysecond, sixtythird, sixtyfourth, sixtyfifth, fiftysixth; sixtyseventh, sixtyeighth, sixtyninth, seventieth, seventyfirst, seventysecond; seventythird, seventyfourth, seventyfifth, seventysixth, seventyseventh, seventyeighth, sixtysixth; eightieth, eightyfirst, eightysecond, eightythird, eightyfourth, eightyfifth, eightysixth, eightyseventh, seventyninth; eightyninth, eightyeighth, ninetyfirst, ninetysecond, ninetythird, ninetyfourth, ninetyfifth, ninetysixth, ninetyseventh, ninetyeighth, ninetyninth, onehundredth; ...

In case you are still wondering what this is all about, the sequence of written-out ordinals reproduces itself if you take its 9th, 20th, 30th, 4th, 90th; 6th, 7th, 8th, 1st, 10th, 2nd, 12th, 15th, 38th; 13th, etc. letters. I have replaced commas with semicolons where the spelling of each ordinal ends. The semicolon after "onehundredth" indicates the end of "thirtyeighth".

So I told Éric that there were 24 swaps, i.e. red-lettered ordinals (where the number does not correspond to its position in the sequence). But in my mind "swap" has a strong sense of pairwise exchanges, which is not the case here. There are three such pairwise exchanges: Positions 1 with 9, 3 with 30, and 13 with 15. But the rest are more involved. Position 14 goes to 38, which goes to 23 (which goes to 14). Position 2 goes to 20, which goes to 17, which goes to 11 (which goes to 2). Finally, position 5 goes to 90, which goes to 88, which goes to 79, which goes to 66, which goes to 56, which goes to 54, which goes to 50, which goes to 43, which goes to 29, which goes to 16 (which goes to 5).

Update: I've calculated 11001 indexed terms for

S10 = tenth, eighteenth, twentyeighth, ninetieth, fifth; second, seventh, eighth, ninth, first, sixth, eleventh, eleventhousandth, fourth, fifteenth; fourteenth, seventeenth, twelfth, third, sixteenth, twentyfirst, twentysecond, twentythird, twentyfourth, twentyfifth, twentieth, twentyseventh; ...

In the linked file, the 1098 red entries are indicated by an asterisk at the end of the line.