A glass clearly

In The Annotated Sherlock Holmes, volume II, pages 332-333 (The Adventure of the Empty House), sidenote 15 lends credence to Sherburne Tupper Bigelow's March 1961 observation (Baker Street Journal: Those Five Volumes) on "Maybe you collect yourself, sir."

The old bookseller now mentions three titles. Then he says: "With five volumes you could just fill that gap on the second shelf." Obviously, two of the three titles must have been two-volume editions or one of the three titles must have been a three-volume edition ...

There is nothing obvious about it. Slightly earlier, the book collector was noted as having wedged under his right arm at least a dozen "precious volumes". He says: "Maybe you collect yourself, sir; here's British Birds, and Catullus, and The Holy War — a bargain every one of them. With five volumes you could just fill that gap on the second shelf."

A "volume" is a book that is part of a larger collection. In a very narrow sense, that larger collection may have the same title with individual books numbered, as with The Annotated Sherlock Holmes. However, the word was never (nor is now) always all that focused. The bookseller has at least a dozen volumes under his right arm: it's just another word for books. Connecting the five volumes to the three titles is the intervening "a bargain every one of them." The bookseller points to the three titles, then sweeps that hand over the rest of the books as he delivers the hawkers' mantra. Five volumes of this stack will fill the gap.

Proportionality

In the list of consecutive positive base-ten integers: 1, 2, 3, ...

   the 1111111111st occurrence of 1 is the first 1 in 1111111111,

   the 2222222222nd occurrence of 2 is the second 2 in 2222222222,

   the 3333333333rd occurrence of 3 is the third 3 in 3333333333,

   the 4444444444th occurrence of 4 is the fourth 4 in 4444444444,

   the 5555555555th occurrence of 5 is the fifth 5 in 5555555555,

   the 6666666666th occurrence of 6 is the sixth 6 in 6666666666,

   the 7777777777th occurrence of 7 is the seventh 7 in 7777777777,

   the 8888888888th occurrence of 8 is the eighth 8 in 8888888888, and

   the 9999999999th occurrence of 9 is the ninth 9 in 9999999999.

In other bases, the n-th occurrence of 1 as the first 1 in n is given in A023037 as a base-ten equivalent. Be careful about the offset (base two is the third term). Multiply the number by d for the n-th occurrence of d as the d-th d in n, up to d = base - 1. These n-th occurrence in n assertions are special solutions to the more general problem but they are not the only ones.

On target (revisited)

one

two

three

eleven

twenty-three

twenty-four

twenty-nine

thirty-one

one hundred eight

one hundred nine

one hundred ninety-eight

one hundred ninety-nine

two hundred forty

two hundred forty-one

two hundred forty-three

two hundred forty-four

two hundred forty-five

two hundred forty-six

two hundred forty-seven

two hundred forty-eight

two hundred forty-nine

two hundred fifty

two hundred fifty-one

four hundred fifty-three

four hundred fifty-four

five hundred fifty-nine

one thousand, one hundred seventy-four

one thousand, seven hundred sixteen

five thousand, five hundred fifty-six

five thousand, five hundred fifty-seven

six thousand, nine hundred fifty-six

six thousand, nine hundred fifty-seven

fifteen thousand, seven hundred fifty-six

seventeen thousand, one hundred fifty-five

twenty-four thousand, nine hundred ninety-eight

twenty-four thousand, nine hundred ninety-nine

forty-three thousand, five hundred sixty-eight

forty-three thousand, five hundred sixty-nine

seven hundred thirty-five thousand, seven hundred fifty-nine

one million, one hundred five thousand, eight hundred five

one million, one hundred five thousand, eight hundred six

one million, one hundred five thousand, eight hundred seven

one million, one hundred seven thousand, seven hundred eighty-four

one million, one hundred seven thousand, seven hundred eighty-five

one million, five hundred eighty-four thousand, five hundred three

one million, five hundred eighty-four thousand, five hundred four

one million, seven hundred seven thousand, nine hundred forty

one million, seven hundred seven thousand, nine hundred forty-one

one million, nine hundred twenty-one thousand, five hundred sixty-six

one million, nine hundred twenty-one thousand, five hundred sixty-seven

two million, two hundred seventeen thousand, seven hundred nine

two million, two hundred seventeen thousand, seven hundred ten

two million, five hundred sixty-eight thousand, seven hundred one

two million, five hundred sixty-eight thousand, seven hundred two

three million, five hundred forty-one thousand, two hundred seventy-nine

three million, five hundred forty-one thousand, two hundred eighty

three million, five hundred forty-one thousand, two hundred eighty-one

three million, five hundred forty-one thousand, two hundred eighty-two

three million, five hundred forty-one thousand, two hundred eighty-three

three million, five hundred forty-one thousand, two hundred eighty-four

three million, five hundred forty-one thousand, two hundred eighty-five

four million, five hundred twelve thousand, five hundred ninety-six

four million, five hundred twelve thousand, five hundred ninety-seven

twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-seven

two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-eight

five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

two billion, two hundred twenty-two million, two hundred twenty-two thousand, two hundred twenty-two

six billion, six hundred sixty-six million, six hundred sixty-six thousand, six hundred sixty-six

eight billion, eight hundred eighty-eight million, eight hundred eighty-eight thousand, eight hundred eighty-eight

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-one

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-two

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-three

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-four

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-five

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-six

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-six

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-seven

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-eight

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, eight hundred eighty-two million, five hundred thirty-six thousand, one hundred seventy-one

nine billion, eight hundred eighty-two million, five hundred thirty-six thousand, one hundred seventy-two

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-one

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-two

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-three

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-four

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-five

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-six

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-eight

ten billion, twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

ten billion, twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

ten billion, two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-seven

ten billion, two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-eight

ten billion, five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

ten billion, five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

twelve billion, two hundred twenty-two million, two hundred twenty-two thousand, two hundred twenty-two

sixteen billion, six hundred sixty-six million, six hundred sixty-six thousand, six hundred sixty-six

eighteen billion, eight hundred eighty-eight million, eight hundred eighty-eight thousand, eight hundred eighty-eight

...

Breakout

The red polygon atop a Google map (click on it for a better view) represents the 15 hectare confines of my time in south Weston / north Mount Dennis since March 20, with the sole exception of a trip to the vet on June 10. Yesterday, having finally received some cloth masks via Amazon, I decided to break out and walked to the yellow open-circle spot to the northwest, where I took this photo of a condo construction site:

Root words (revisited)

A few days ago, Futility Closet highlighted a David Morice "Kickshaws" bit (from a 1997 "Word Ways") dealing with "root words" wherein the number of letters of a specific power of an integer written in English words is equal to that integer. This looked like something I could verify and possibly expand on. In short order, Mathematica came up with over one hundred examples. I trust of course that Mathematica is correctly providing the (American) English equivalents of the large numbers. Also, Mathematica does not ever include the word "and" in its integer wording. The solutions list begins:

3^0 = 1: one

4^1 = 4: four

9^2 = 81: eighty-one

10^2 = 100: one hundred

10^6 = 1000000: one million

10^7 = 10000000: ten million

10^9 = 1000000000: one billion

10^10 = 10000000000: ten billion

34^3 = 39304: thirty-nine thousand, three hundred four

50^7 = 781250000000: seven hundred eighty-one billion, two hundred fifty million

57^5 = 601692057: six hundred one million, six hundred ninety-two thousand, fifty-seven

60^9 = 10077696000000000: ten quadrillion, seventy-seven trillion, six hundred ninety-six billion

66^4 = 18974736: eighteen million, nine hundred seventy-four thousand, seven hundred thirty-six

70^8 = 576480100000000: five hundred seventy-six trillion, four hundred eighty billion, one hundred million

82^6 = 304006671424: three hundred four billion, six million, six hundred seventy-one thousand, four hundred twenty-four

84^5 = 4182119424: four billion, one hundred eighty-two million, one hundred nineteen thousand, four hundred twenty-four

88^5 = 5277319168: five billion, two hundred seventy-seven million, three hundred nineteen thousand, one hundred sixty-eight

88^6 = 464404086784: four hundred sixty-four billion, four hundred four million, eighty-six thousand, seven hundred eighty-four

In addition to the five powers of ten, 88 has both a fifth and a sixth power that qualify. Another integer with multiple powers (59 and 60) is 2326. My final entry (the full list is here):

3411^83 = 169505077051870754738656270298647848404856613833009765867153972878059706815231408155486012246885256492743555141825746171932166403032624146125802858179156124954911215966912803668763183796090700586141654608753614304795644365897817582455004645907480837144225881607352543409279188776048394771808331: one hundred sixty-nine senonagintillion, five hundred five quinonagintillion, seventy-seven quattuornonagintillion, fifty-one trenonagintillion, eight hundred seventy duononagintillion, seven hundred fifty-four unonagintillion, seven hundred thirty-eight nonagintillion, six hundred fifty-six novoctogintillion, two hundred seventy octoctogintillion, two hundred ninety-eight septenoctogintillion, six hundred forty-seven sexoctogintillion, eight hundred forty-eight quintoctogintillion, four hundred four quattuoroctogintillion, eight hundred fifty-six tresoctogintillion, six hundred thirteen duooctogintillion, eight hundred thirty-three unoctogintillion, nine octogintillion, seven hundred sixty-five novemseptuagintillion, eight hundred sixty-seven octoseptuagintillion, one hundred fifty-three septenseptuagintillion, nine hundred seventy-two seseptuagintillion, eight hundred seventy-eight quinseptuagintillion, fifty-nine quattuorseptuagintillion, seven hundred six treseptuagintillion, eight hundred fifteen duoseptuagintillion, two hundred thirty-one unseptuagintillion, four hundred eight septuagintillion, one hundred fifty-five novemsexagintillion, four hundred eighty-six octosexagintillion, twelve septensexagintillion, two hundred forty-six sesexagintillion, eight hundred eighty-five quinsexagintillion, two hundred fifty-six quattuorsexagintillion, four hundred ninety-two tresexagintillion, seven hundred forty-three duosexagintillion, five hundred fifty-five unsexagintillion, one hundred forty-one sexagintillion, eight hundred twenty-five novemquinquagintillion, seven hundred forty-six octoquinquagintillion, one hundred seventy-one septenquinquagintillion, nine hundred thirty-two sexquinquagintillion, one hundred sixty-six quinquinquagintillion, four hundred three quattuorquinquagintillion, thirty-two tresquinquagintillion, six hundred twenty-four duoquinquagintillion, one hundred forty-six unquinquagintillion, one hundred twenty-five quinquagintillion, eight hundred two novemquadragintillion, eight hundred fifty-eight octoquadragintillion, one hundred seventy-nine septenquadragintillion, one hundred fifty-six sexquadragintillion, one hundred twenty-four quinquadragintillion, nine hundred fifty-four quattuorquadragintillion, nine hundred eleven tresquadragintillion, two hundred fifteen duoquadragintillion, nine hundred sixty-six unquadragintillion, nine hundred twelve quadragintillion, eight hundred three novemtrigintillion, six hundred sixty-eight octotrigintillion, seven hundred sixty-three septrigintillion, one hundred eighty-three sextrigintillion, seven hundred ninety-six quintrigintillion, ninety quattuortrigintillion, seven hundred trestrigintillion, five hundred eighty-six duotrigintillion, one hundred forty-one untrigintillion, six hundred fifty-four trigintillion, six hundred eight novemvigintillion, seven hundred fifty-three octovigintillion, six hundred fourteen septenvigintillion, three hundred four sexvigintillion, seven hundred ninety-five quinvigintillion, six hundred forty-four quattuorvigintillion, three hundred sixty-five trevigintillion, eight hundred ninety-seven duovigintillion, eight hundred seventeen unvigintillion, five hundred eighty-two vigintillion, four hundred fifty-five novemdecillion, four octodecillion, six hundred forty-five septendecillion, nine hundred seven sexdecillion, four hundred eighty quindecillion, eight hundred thirty-seven quattuordecillion, one hundred forty-four tredecillion, two hundred twenty-five duodecillion, eight hundred eighty-one undecillion, six hundred seven decillion, three hundred fifty-two nonillion, five hundred forty-three octillion, four hundred nine septillion, two hundred seventy-nine sextillion, one hundred eighty-eight quintillion, seven hundred seventy-six quadrillion, forty-eight trillion, three hundred ninety-four billion, seven hundred seventy-one million, eight hundred eight thousand, three hundred thirty-one

This is likely the largest solution you will see for a while. Mathematica's large-integer naming system is relatively recent and when it was introduced a few years ago it arrived with a bug. Although that "-illions" bug was eventually fixed, the naming of integers 10^306 and larger remains essentially aberrant/unusable. If you are curious about how many times each letter appears in the preceding:

{a,68}, {b,1}, {c,29}, {d,209}, {e,369}, {f,86}, {g,113}, {h,148}, {i,441}, {l,195}, {m,8}, {n,426}, {o,238}, {p,19}, {q,50}, {r,184}, {s,108}, {t,307}, {u,197}, {v,69}, {w,22}, {x,50}, {y,74}.

There are, in addition, 371 spaces, 97 commas, and 70 hyphens. If you are a user of Mathematica, I will alert you to the fact that the hyphen it uses for its integer names is perversely different from the keyboard hyphen that I have used here.

My largest Leyland prime find

Late yesterday, I found my (to-date) largest Leyland prime: 33845^26604+26604^33845. At 149763 decimal digits, this becomes for the moment the fifth largest known such prime:

386434  (328574,15)      Serge Batalov        May 2014
300337  (314738,9)       Anatoly Selevich     Feb 2011
265999  (255426,11)      Serge Batalov        May 2014
223463  (234178,9)       Anatoly Selevich     Jul 2011
149763   (33845,26604)   Hans Havermann       Sep 2020

The number is the 167th new Leyland prime discovered since I (using xyyxsieve and pfgw) started finding them two months ago. Prior to that I had found 579 new Leyland primes using Mathematica — but that took from 3 October 2015 to 3 July 2020. At my current rate of discovery, I will find my 1000th new Leyland prime on December 9, but that is likely early because I am entering large-number terrain where my finds will be slower in coming. Still, I might have it by the end of the calendar year. We'll see.

Visually impaired

Going behind the garden shed just prior to sunset, August 25, I had a smallish raccoon approach me atop the neighbour's back fence. It seemed not to notice me until fairly close, whereupon it retreated to that yard's maple tree. I ran in the house to fetch my camera. It was only after looking at the photos just now that I noticed the eyes.

Display bug

 

I have had insects walk on my iMac display screen before but this one is behind the glass.

Boo!

The eyes have it.

The iMac from hell

I had mentioned in my previous post that I was considering purchasing a new computer "if I can trade in my old kernel-panic-plagued late-2015 one". When I bought this iMac it was pretty much top of the line.

SSDs were expensive and 1 TB was as big as Apple was willing to provide at the time. The standard 8 GB RAM was going to be replaced with 64 GB purchased from Other World Computing (Apple has always been incredibly ungenerous with its memory pricing).

I experienced my first kernel panic on the new iMac on 23 December 2015, 15 days after receipt. There was another one on 9 February 2016. I seem to have been panic-free until May 8 and then more on May 14, May 21, and May 22, when I decided to do a RAM switch. In addition to the kernel panics there was other weirdness happening in my running Mathematica Leyland prime searches. For example, the Mathematica front end would lose its connection with the Mathematica kernel. Mathematica was pretty much the only thing I was running on this machine.

I tried an OS reinstall on June 13 but the situation did not improve. I should of course have brought it back to Apple before its one-year warranty ran out but I was still under the impression that it might be the third party RAM and, besides, Catherine was increasingly anxious about driving and I did not ask her for the ride.

By July 2017 I had boxed the iMac from hell and purchased a new one, the one that I am currently working on. It's been fine but the 2 TB SSD is getting full. A 4 TB SSD in a new iMac should do the trick for a few more years. Plus I can upgrade to 128 GB RAM. The idea of trading in the defective iMac seemed like a good way to reduce the cost.

Phobio is Apple's official Mac Trade In partner for U.S and Canada. When I tried to evaluate nine days ago my late-2015 iMac on Phobio's website (by entering the computer's serial number), I got this:

My device (iMac Core i7 4.0 GHz) did not appear! Selecting one of the Core i5 alternatives instead would of course end up in my iMac being undervalued and this was not fair. So I sent an email (August 14) to Phobio explaining my dilemma. No response. A followup email (August 19) noting the non-response and asking for them to "please let me know" has also not gotten a reply. At this point I am entertaining deep-Apple conspiracy theories about what is going on (and I'm not the sort of person who entertains conspiracy theories).

Strangely, my late-2014 iMac which has never had kernel panics before started to experience them in June 2019. At any rate, I will give up for the moment the idea of trading in the late-2015 machine. I'll likely still purchase a new iMac but am unsure as to when.