Counting digits (reprise, see previous post: Spam)

1

199981

199982

199983

199984

199985

199986

199987

199988

199989

199990

199991

200001

1599981

1599982

1599983

1599984

1599985

1599986

1599987

1599988

1599989

1599990

1599991

2600001

13199998

13199999

28263827

28263828

35000001

35199981

35199982

35199983

35199984

35199985

35199986

35199987

35199988

35199989

35199990

35199991

35200001

117463825

117463826

242463827

242463828

371599983

371599984

371599985

371599986

371599987

371599988

371599989

371599990

371599991

371599992

371599993

499999984

499999985

499999986

499999987

499999988

499999989

499999990

499999991

499999992

499999993

499999994

500000001

500199981

500199982

500199983

500199984

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500200001

501599981

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501599990

501599991

502600001

513199998

513199999

528263827

528263828

535000001

535199981

535199982

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535199984

535199985

535199986

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535199988

535199989

535199990

535199991

535200001

1111111110

1111111111

2222222222

3333333333

4444444444

5555555555

6666666666

7777777777

8888888888

9471736170

9471736171

9486799989

9486799990

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9486799993

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9498399989

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9628399986

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9757536170

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9882536171

9882536172

9971736170

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9986799989

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9998399989

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9998399996

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9999999999

10028263827

10028263828

10242463827

10242463828

10371599983

10371599984

10371599985

10371599986

10371599987

10371599988

10371599989

10371599990

10371599991

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10371599993

10499999984

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10499999986

10499999987

10499999988

10499999989

10499999990

10499999991

10499999992

10499999993

10499999994

10528263827

10528263828

12222222222

13333333333

14444444444

15555555555

16666666666

17777777777

18888888888

19471736170

19471736171

19486799989

19486799990

19486799991

19486799992

19486799993

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19486799995

19486799996

19486799997

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19498399989

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19498399991

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19628399986

19628399987

19628399988

19628399989

19628399990

19628399991

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19628399993

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19628399995

19628399996

19757536170

19757536171

19882536171

19882536172

19971736170

19971736171

19986799989

19986799990

19986799991

19986799992

19986799993

19986799994

19986799995

19986799996

19986799997

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19998399989

19998399990

19998399991

19998399992

19998399993

19998399994

19998399995

19998399996

19998399997

19998399998

19999999999

20371599983

20371599984

20371599985

20371599986

20371599987

20371599988

20371599989

20371599990

20371599991

20371599992

20371599993

20499999984

20499999985

20499999986

20499999987

20499999988

20499999989

20499999990

20499999991

20499999992

20499999993

20499999994

23333333333

24444444444

25555555555

26666666666

27777777777

28888888888

29471736170

29471736171

29486799989

29486799990

29486799991

29486799992

29486799993

29486799994

29486799995

29486799996

29486799997

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29498399989

29498399990

29498399991

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29498399993

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29498399995

29498399996

29498399997

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29628399986

29628399987

29628399988

29628399989

29628399990

29628399991

29628399992

29628399993

29628399994

29628399995

29628399996

29757536170

29757536171

29882536171

29882536172

29971736170

29971736171

29986799989

29986799990

29986799991

29986799992

29986799993

29986799994

29986799995

29986799996

29986799997

29986799998

29998399989

29998399990

29998399991

29998399992

29998399993

29998399994

29998399995

29998399996

29998399997

29998399998

29999999999

30499999984

30499999985

30499999986

30499999987

30499999988

30499999989

30499999990

30499999991

30499999992

30499999993

30499999994

34444444444

35555555555

36666666666

37777777777

38888888888

39471736170

39471736171

39486799989

39486799990

39486799991

39486799992

39486799993

39486799994

39486799995

39486799996

39486799997

39486799998

39498399989

39498399990

39498399991

39498399992

39498399993

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39498399995

39498399996

39498399997

39498399998

39628399986

39628399987

39628399988

39628399989

39628399990

39628399991

39628399992

39628399993

39628399994

39628399995

39628399996

39757536170

39757536171

39882536171

39882536172

39971736170

39971736171

39986799989

39986799990

39986799991

39986799992

39986799993

39986799994

39986799995

39986799996

39986799997

39986799998

39998399989

39998399990

39998399991

39998399992

39998399993

39998399994

39998399995

39998399996

39998399997

39998399998

39999999999

45555555555

46666666666

47777777777

48888888888

49471736170

49471736171

49486799989

49486799990

49486799991

49486799992

49486799993

49486799994

49486799995

49486799996

49486799997

49486799998

49498399989

49498399990

49498399991

49498399992

49498399993

49498399994

49498399995

49498399996

49498399997

49498399998

49628399986

49628399987

49628399988

49628399989

49628399990

49628399991

49628399992

49628399993

49628399994

49628399995

49628399996

49757536170

49757536171

49882536171

49882536172

49971736170

49971736171

49986799989

49986799990

49986799991

49986799992

49986799993

49986799994

49986799995

49986799996

49986799997

49986799998

49998399989

49998399990

49998399991

49998399992

49998399993

49998399994

49998399995

49998399996

49998399997

49998399998

49999999999

56666666666

57777777777

58888888888

59471736170

59471736171

59486799989

59486799990

59486799991

59486799992

59486799993

59486799994

59486799995

59486799996

59486799997

59486799998

59498399989

59498399990

59498399991

59498399992

59498399993

59498399994

59498399995

59498399996

59498399997

59498399998

59628399986

59628399987

59628399988

59628399989

59628399990

59628399991

59628399992

59628399993

59628399994

59628399995

59628399996

59757536170

59757536171

59882536171

59882536172

59971736170

59971736171

59986799989

59986799990

59986799991

59986799992

59986799993

59986799994

59986799995

59986799996

59986799997

59986799998

59998399989

59998399990

59998399991

59998399992

59998399993

59998399994

59998399995

59998399996

59998399997

59998399998

59999999999

67777777777

68888888888

69471736170

69471736171

69486799989

69486799990

69486799991

69486799992

69486799993

69486799994

69486799995

69486799996

69486799997

69486799998

69498399989

69498399990

69498399991

69498399992

69498399993

69498399994

69498399995

69498399996

69498399997

69498399998

69757536170

69757536171

69882536171

69882536172

69971736170

69971736171

69986799989

69986799990

69986799991

69986799992

69986799993

69986799994

69986799995

69986799996

69986799997

69986799998

69998399989

69998399990

69998399991

69998399992

69998399993

69998399994

69998399995

69998399996

69998399997

69998399998

69999999999

78888888888

79486799989

79486799990

79486799991

79486799992

79486799993

79486799994

79486799995

79486799996

79486799997

79486799998

79498399989

79498399990

79498399991

79498399992

79498399993

79498399994

79498399995

79498399996

79498399997

79498399998

79882536171

79882536172

79986799989

79986799990

79986799991

79986799992

79986799993

79986799994

79986799995

79986799996

79986799997

79986799998

79998399989

79998399990

79998399991

79998399992

79998399993

79998399994

79998399995

79998399996

79998399997

79998399998

79999999999

89999999999

100559404366

Spam

On 18 June 2022, Blogger informed me that my list of numbers in a 2020 "Counting digits" blog had been flagged for review. The Blogger Team determined that it violated their guidelines (specifically, their Spam policy) and unpublished the URL. That was unfortunate because my post was a link in OEIS sequence A339752. They allowed me to update the content (so as to adhere to the guidelines) and republish, which would trigger a review. This was for me problematic, as I did not think that my numbers-list violated their guidelines. I decided to add an introduction:

On 7 October 2020, I wrote a little thing called Proportionality wherein I presented nine special solutions to OEIS sequence A339752: "Base-ten n containing a digit making its n-th appearance in the list of consecutive positive integers." The full sequence contains 662 terms. Wouldn't it be neat (I thought) if I showed those 662 terms in a manner where the digit making its n-th appearance stood out from the rest of the digits of the term.

I also added a postscript explaining what had happened and why I had previously published the numbers without an introduction, "leaving it to my audience to discover the sequence and marvel at the placement of the special digits". I then republished and waited.

Yesterday, Blogger informed me that "Counting digits" had been re-evaluated and reinstated. Did the introduction turn the spam into a now-understandable work of art? Did they realize that there was never any reason to unpublish the post in the first place? I did not know. At any rate, since the introduction and postscript are now documented herein, I decided to remove them from my original post. Bad mistake! As soon as I updated the blog I received an email repeating their original Spam claim and unpublishing the URL. This time I just deleted it.

Update: After a night's sleep, re-creation.

Exceptional fractions

I am assuming that you are acquainted with divisors and know how to calculate and count them. I want to introduce here the concept of exceptional fractions:

4/5, 4/7, 6/7, 4/11, 6/11, 36/143, 8/11, 9/11, 6/13, 8/11, 9/11, ...

The use of "exceptional" comes from OEIS A072066 wherein are defined "exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m)". The fractions are in fact the reduced ratios A005179(m)/A037019(m) which are therefore numbers between zero and one. There is one such fraction for each term in A072066, so a better approach to presenting these fractions is to couple them with those terms:

  8 4/5
 16 4/7
 24 6/7
 32 4/11
 48 6/11
 64 36/143
 72 8/11
 80 9/11
 96 6/13
108 8/11
112 9/11
...

You will notice that the fraction at 108 repeats the fraction at 72 and that the fraction at 112 repeats the fraction at 80. As we chart the fractions [19 MB] for larger exceptional numbers [to 10^8], an overwhelming majority of them will be such repeats.

Are these fractions good for anything? Absolutely. Let's do a little refresh. A037019 terms are "an easy way to produce a number with exactly n divisors". A005179 terms, on the other hand, are the "smallest number with exactly n divisors". Mostly, the easy way also produces the smallest number. But not always. A072066 terms are of course the exceptions. The easy way of finding a number with exactly 8 divisors produces 30. But the smallest number with exactly 8 divisors is actually 24. What the fraction for the first exception, 8, allows one to do is determine the 24 as 4/5 of that 30. This becomes increasingly useful as our exceptions increase in size. What is the smallest number with exactly 3603600 divisors? Looking at our chart, 3603600 isn't even in there as an exceptional number, so the easy calculation, which produces 2549066103582535692163008000000, also produces the smallest number. What is the smallest number with exactly 10000000 divisors? This time, 10000000 is in the chart alongside 32/43. Our easy calculation produces 1740652905587144828469399739530000 and multiplying this by 32/43 yields 1295369604157875221186530038720000. This is the smallest number.

My most common exceptional fractions are 9/11, 9/13, 9/17, 15/17, 9/19, 25/29, 15/19, 9/23, 25/31, 15/23, 9/29, 27/29, 21/23, 25/37, 15/29, 35/37, 225/899, 27/31, 21/29, 25/41, ...

Here are the exceptional fractions for small powers of two:

2^  3 4/5
2^  4 4/7
2^  5 4/11
2^  6 36/143
2^  7 36/221
2^  8 36/323
2^  9 576/7429
2^ 10 576/12673
2^ 11 576/20677
2^ 12 576/33263
2^ 13 14400/1363783
2^ 14 14400/2022161
2^ 15 14400/3065857
2^ 16 14400/4391633
2^ 17 14400/6319667
2^ 18 14400/8965109
2^ 19 14400/12780049
2^ 20 705600/907383479
2^ 21 705600/1249792339
2^ 22 705600/1673450759
2^ 23 705600/2276990377
2^ 24 705600/3024658859
2^ 25 705600/4132280413
2^ 26 705600/5717264681
2^ 27 705600/7454155217
2^ 28 57153600/797594608219
2^ 29 57153600/1047443521637
2^ 30 57153600/1329900201629
2^ 31 57153600/1741209542339
2^ 32 57153600/2258400495509
2^ 33 57153600/3003891921211
2^ 34 57153600/3902252121947
2^ 35 57153600/5334271249267
2^ 36 57153600/7128096979109
2^ 37 6915585600/1119111225720113
2^ 38 6915585600/1436339604664397
2^ 39 6915585600/1831058885335529
2^ 40 6915585600/2312213044985741
2^ 41 6915585600/2977598093902501
2^ 42 6915585600/3617082248297669
2^ 43 6915585600/4575249731290429
2^ 44 6915585600/5624351580503521
2^ 45 6915585600/6797529210792599
2^ 46 6915585600/8100049778130103
2^ 47 1168733966400/1709110503185451733
2^ 48 1168733966400/2203073076360437783
2^ 49 1168733966400/2793841275607929479
2^ 50 1168733966400/3534749459194562711
2^ 51 1168733966400/4312024209383942993
2^ 52 1168733966400/5339760549444364639
2^ 53 1168733966400/6532397423431938467
2^ 54 1168733966400/8239355544127721383
2^ 55 1168733966400/10035613150904381021
2^ 56 1168733966400/11835723133129382101
2^ 57 1168733966400/14025592611505743547
2^ 58 1168733966400/16597972042436927953
2^ 59 1168733966400/19732353028991540957
2^ 60 1168733966400/23199963184713903803
2^ 61 1168733966400/27243110295742882889
2^ 62 1168733966400/31801718393038504727
2^ 63 299195895398400/9763127546662820951189
2^ 64 299195895398400/11814523996156176326147
2^ 65 299195895398400/14060631219760012129597
2^ 66 299195895398400/16569591437412356301421
2^ 67 299195895398400/20238135667097748840481
2^ 68 299195895398400/24621847363942026567661
2^ 69 299195895398400/30404914716326986544407
2^ 70 299195895398400/37495813554763668918721
2^ 71 86467613770137600/13236022184831575128308513
2^ 72 86467613770137600/16217515236704899218644219
2^ 73 86467613770137600/19387062188503902323265239
2^ 74 86467613770137600/23252007062096320149768277
2^ 75 86467613770137600/28154986187011199158984591
2^ 76 86467613770137600/34016907601341606554861509
2^ 77 86467613770137600/39977574190096329153598571
2^ 78 86467613770137600/47095243185365705264031551
2^ 79 86467613770137600/54424185928909647869961533
2^ 80 86467613770137600/63780779498349701945026553
2^ 81 86467613770137600/75705797761497238286023019
2^ 82 86467613770137600/88780336650669463282494961
2^ 83 31214808571019673600/38264325096438538674755328191
2^ 84 31214808571019673600/45145647865825305847872090209
2^ 85 31214808571019673600/53133885825998148169479484187
2^ 86 31214808571019673600/62106362588171977939523513179
2^ 87 31214808571019673600/72808764496316496331190750437
2^ 88 31214808571019673600/85536260603641745047182963881
2^ 89 31214808571019673600/99325481456621774475444197353
2^ 90 31214808571019673600/114682538439939854319527838839
2^ 91 31214808571019673600/130945587900860420457749390557
2^ 92 31214808571019673600/149696746072821339855040472737
2^ 93 31214808571019673600/173164644507040362255118076539
2^ 94 31214808571019673600/197271091538182872081816648679
2^ 95 31214808571019673600/227340126276104510782509255637
2^ 96 31214808571019673600/260483105961003573857863680149
2^ 97 31214808571019673600/299290972763320133394249691187
2^ 98 31214808571019673600/347284179977037393092214006923
2^ 99 31214808571019673600/397439006844618285748857605297
2^100 31214808571019673600/466408899572534691084884955457
2^101 31214808571019673600/551027360834074462253632981933
2^102 31214808571019673600/657221070630791168041271029843
2^103 31214808571019673600/772474870073351623397151544471
2^104 31214808571019673600/902542507334162369020491229577
2^105 31214808571019673600/1049596276349911838514664953337
2^106 31214808571019673600/1213661425759316895436796950051
2^107 31214808571019673600/1416340471015345959485884313479
2^108 31214808571019673600/1650078387646562188556246361283
2^109 31214808571019673600/1897115075240481287802670960477
2^110 31214808571019673600/2180050019540208898603069306399
2^111 16512633734069407334400/1323290361860906801452063068984193
2^112 16512633734069407334400/1499402942367349111441986434911849
2^113 16512633734069407334400/1691282660769020844167651975028539
2^114 16512633734069407334400/1879540335755877742441250579071213
2^115 16512633734069407334400/2106554088564758180249430045104681
2^116 16512633734069407334400/2373112778154674856836352651866609
2^117 16512633734069407334400/2672349415680308113740411129860297
2^118 16512633734069407334400/2996551251204782234991414213205567
2^119 16512633734069407334400/3333471834815541395995559593225273
2^120 16512633734069407334400/3704482190798384114605520694663499
2^121 16512633734069407334400/4087917743101388814280883437683761
2^122 16512633734069407334400/4577651649096896292863618225559353
2^123 16512633734069407334400/5105552168762106738498631859478883
2^124 16512633734069407334400/5688567913971482711899780685194253
2^125 16512633734069407334400/6370827274804367186260532339496319
2^126 16512633734069407334400/7214781776474735698818470387701001

This suggests that we can get as close to zero as we like. The numerators are squares and the denominators appear to be the products of consecutive primes. As for approaching one (in the other direction), my largest exceptional fraction is currently 900/901 at 4354560 (thus beating 400/403 at 64800).

Offspring

I mentioned last month that a pair of robins had been making daily forages around our yard since at least mid-April. This young specimen spotted today must surely be the outcome:

Spellbound

Leonard Susskind's recent lecture* on the quantum origins of gravity was for me something of a revelation. I hadn't any idea that physicists had to a certain extent reconciled their notion of gravity with that of quantum mechanics. And I hadn't been particularly aware of Susskind as a physics celebrity either. He is even known for things that he has said!

video @18:21

I can't watch stuff that has written text without doing a bit of a spell-check. At 18:21 Susskind had written "things are not where you think the are" which he then repeated exactly at 19:40. The word "they" doesn't strike me as especially vulnerable to being typoed. Somewhat more comprehensible is the spelling at 50:28 of "communication" by the dropping of an m.

video @19:40

* Update: A synopsis of  Susskind's lecture appears as an 18 June 2022 Science Time video.

What's a heaven for?

The west-wall, floor-to-ceiling bookshelf in my room is compartmentalized into 62 sections, of which 59 are used for books. The picture is a composite of three of those sections. The upper shown section is on the left side of the bookshelf, one down from the top; the bottom two, horizontally adjacent at the top-right corner of the shelf. Most of these books are from Springer and were purchased in the book-buying mania of my 1980s (give or take).

Five of the books in the middle shown section are Berger's Geometry I and II, Weil's Basic Number Theory, Armstrong's Groups and Symmetry, and Gelbaum & Olmsted's Theorems and Counterexamples in Mathematics. They are all pretty much beyond my grasp and I was fortunate enough yesterday to be allowed to ship them to a mathematician who might actually benefit from their content.

A million-digit Leyland prime (reality check)

Having previously decided that finding a million-digit Leyland prime was doable and subsequently committing to a search thereof, it is time for a reality check.

The notion that I could scan the Leyland interval from L(999999,10) to L(1000099,10) by the end of September was predicated on an estimate of about 5 hours examination time per candidate (of some 59500 candidates). It turns out that the running time on my nine oldest Mac minis is closer to 11.5 hours and, for reasons I don't comprehend, three newer Mac minis are running at 18.5 hours per candidate. My 2020 iMacs, which I had assumed would be the most productive, still need 11 hours on my main machine (7 processes) and almost 15 hours on my helper machine (8 processes). I'm well into 2023 at this rate!

Early this morning Gabor Levai noted that he had discovered a 1000027-digit Leyland prime. Of my 108 running processes, this will have been found in about four months (by process #23 that happens to reside on one of the slower Mac minis). Disheartening, to say the least.

Aerial lies

my neighbourhood, looking toward downtown Toronto

The blue arrow at bottom-left points to a lot that seemed to me unfamiliar, as it is the location of what I had always considered a shallow pit waiting on the construction of a nine-storey building (1681 Weston Road). I most certainly was not conscious of a circular path therein. The drone image (taken by a realtor for an area house) appears filtered for effect and my inclination was to think that the lot might have been photoshopped. But I was wrong. I went there to take this over-the-fence photograph:

path in the metre-deep pit at 1681 Weston Road

Thirty

I turned on the air conditioning yesterday. In spite of that, my room registers at 30º — significantly above the ambient air temperature outside — because there are sixteen computers herein generating heat, 24/7, and there is insufficient air flow from the vents to make much of a difference. The living room is a little better (the ceiling fan helps) but even Bodie has taken to lying on the floor instead of his chair or the bed/couch. I expect that it will be like this most of the summer!