10*9*8*7/6/5*4*3-2+1

Happy New Year!

Pleasantly nonplussed

Eric Angelini suggested on Monday the sequence of numbers with the property that if one inserts a single plus anywhere inside them (and executes the additions) only primes result. In deference to the second part of the number-split not starting with it, he only allowed zero as a final digit:

11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 110, 112, 116, 118, 130, 136, 152, 158, 170, ...

Surely this sequence is finite. But what is the largest term? My program hinted that it might be the 11-digit 46884486265 (term #440), since it found no 12-digit representative. But (thankfully) I had the computation plod on:

5391391551358

5 + 391391551358 = 391391551363
53 + 91391551358 = 91391551411
539 + 1391551358 = 1391551897
5391 + 391551358 = 391556749
53913 + 91551358 = 91605271
539139 + 1551358 = 2090497
5391391 + 551358 = 5942749
53913915 + 51358 = 53965273
539139155 + 1358 = 539140513
5391391551 + 358 = 5391391909
53913915513 + 58 = 53913915571
539139155135 + 8 = 539139155143

All prime! By plugging 5391391551358 into Google, I discovered that this number was already known to Giovanni Resta, who treated the sequence from the perspective of allowing internal zeros. The numbers 20, 101, and 1001 appear to be (currently) missing from his magnanimous numbers list. But, he has one larger example:

97393713331910

9 + 7393713331910 = 7393713331919
97 + 393713331910 = 393713332007
973 + 93713331910 = 93713332883
9739 + 3713331910 = 3713341649
97393 + 713331910 = 713429303
973937 + 13331910 = 14305847
9739371 + 3331910 = 13071281
97393713 + 331910 = 97725623
973937133 + 31910 = 973969043
9739371333 + 1910 = 9739373243
97393713331 + 910 = 97393714241
973937133319 + 10 = 973937133329
9739371333191 + 0 = 9739371333191

Nice.

Das Ei des Columbus

Gabriel Fernandes' puzzle blog entry yesterday on the Logic Egg had me thinking about the Richter original called Ei des Columbus, which came out in 1893 to coincide with the 400th anniversary of Christopher Columbus' rediscovery of the Americas. Unbeknownst to me, Columbus' egg was then already a well-worn idiom in Germany, based on an apocryphal account of Columbus shaming his opponents by coming up with something outside the box. In short, Columbus' discovery is belittled by those present at a get-together who suggest that "any of us" could have found the New World. Columbus asks his detractors to stand an egg on its end, something that they are unable to accomplish. So he shows them how by gently hitting the egg onto the table so that its cracked shell now gives it support. Yes, he implores, any of us can do it.

Here is my transcription (I've shortened any long s occurrences but have left the sharp s as is) of the tale from Friedrich Förster's 1842 Christoph Columbus:

     Wir dürfen den Entdecker der neuen Welt nicht zur zweiten Fahrt dahin abreisen lassen, ohne jenes Gastmahles zu gedenken, bei welchem ein sinnreicher Einfall von ihm Veranlassung zu einem Sprüchworte wurde, welches eine eben so weite Verbreitung erhalten hat, wie sein welthistorischer Name; wer hörte nicht von dem Ei des Columbus? Bei einem Feste, welches der Cardinal Mendoza dem Admiral zu Ehren veranstaltete, hielt er ihm eine große Lobrede wegen der von ihm gemachten Endeckung, welche er den größten Sieg nannte, den jemals der Geist eines einzigen Mannes erfochten habe. Die anwesenden Herren vom Hofe nahmen es übel auf, daß einem Ausländer, noch dazu einem Manne, der nicht ein Mal von nobler Herkunst sei, so große Auszeichnung erwiesen würde. "Mich dünkt", bub einer der königlichen Kammerherren an, "der Weg nach der sogenannten neuen Welt war nicht so schwer zu finden, der Ocean stand überall offen und kein Spanischer Seefahrer würde den Weg verfehlt haben." Mit vornehmem Gelächter, hinter welches sich so gern die Dummheit verbirgt, gab die Gesellschaft dieser Aeußerung ihren Beifall zu erkennen und mehrere Stimmem riefen: "o, das hätte ein Jeder von uns gekonnt!"
     "Ich bin weit entfernt", entgegnete Columbus, "mir etwas als Ruhm anzumaßen, was ich nur einer gnädigen Fügung des Himmels zuschreiben darf; indessen kommt es doch bei vielen Dingen in der Welt, welche uns leicht auszuführen scheinen, oft nur darauf an, daß sie ein Anderer uns vormacht. Dürft’ ich", sagte Columbus zu jenem Kammerherrn gewendet, "Ew. Excellenz wohl ersuchen, dies Ei — er hatte sich von einem Diener ein Hühnerei bringen lassen — so auf die Spitze zu stellen, daß es nicht umfällt?" Die Excellenz versuchte von der einen, wie von der anderen Seite vergeblich, daß Ei zum Stehen zu bringen. Der Nachbar hat es sich aus, es gelang ihm eben so wenig; nun drängten sich die Anderen dazu, ein jeder wollte den Preis gewinnen, allein weder mit Eifer noch mit Ruhe war es möglich, das Kunststück auszuführen. "Es ist unmöglich!" riefen die Hidalgos, "Ihr verlangt Unausführbares!" — "Und doch", sagte Columbus, "werden diese Herren sogleich sagen: das kann ein Jeder von uns auch!" Jetzt nahm er das Ei, und setzte es mit einem leichten Schlag auf den Tisch, so daß es auf der eingedrückten Schale fest stand. — "Ja! das kann ein Jeder von uns!" riefen die Hidalgos; seitdem hört man oft sagen, wenn eine glückliche Erfindung gemacht wurde, zu welcher ein jeder sich klug genugt dünkt: "das Ei des Columbus!"

Climb to a prime

I took on my table of A195264 three years ago. I am still chipping away at the currently-317 primary unknowns (out of 10000), extending ECM-factorization attempts from 2000 to 5000 curves. I'm not yet even half-way through the list. Occasionally factordb will have found for me (over time) the large factors for up to 116-digit composites! And in a handful of cases I have evolved an unknown into a prime — which finishes those unknowns from consideration, but there are plenty more.

I've been a little resentful that I hadn't picked the arguably aesthetically-better (one less arithmetic symbol with which to deal), historically better-known home prime sequence (A037274) on which to while away my time. No matter now: John Conway has offered $1000 for a resolution (see the video here, from 19:20 to 22:30) to term #20 of A195264, not term #49 of A037274 — which he could easily have picked instead. I don't believe that (mathematically) it makes any difference: one is as contrived as the other. Conway's point (I think) is that there exist easily-stated but impossible-to-prove conjectures about such sequences. To collect the $1000, you will almost certainly have to luck into a prime in the evolution, because proving that it doesn't evolve into a prime may well be impossible.

While the prize may spur some to further evolve A195265, it may also impede them from sharing any results of that effort. I hope that's not the case.

Time has come

My D5300 has a GPS chip in it. I don't own a mobile phone so this is actually my first acquaintance with the capability. Of course I enabled it, even though I suspect that it will be a drain on the camera's battery. As it turns out, only about one in seven of my photos had the location data embedded in it and a cursory look indicated location placement wasn't necessarily all that close. A handful of waypoints had me on the opposite side of the Humber river from where I actually was!

There is an upside. The camera has a set clock from satellite option and I had turned on this feature. The above photo was captured on September 28, exactly 19 days after my first photo taken just after I manually set the camera's clock. The photo's embedded time-stamp was still in sync with real time, indicated here by a website clock. I may never have to correct the camera clock again.

Shelfie

My new D5300 does some in-camera filter effects. My sense of photography is journalism — not art — so this seems to me extravagant and unnecessary. (If you want to photoshop, use a computer.) Regardless, I tried it out. This "shelfie" is a color sketch effect and its pastel hues are actually quite pleasant. I shared another shelfie on Google+ last year and there is some overlap with this image. The only other effect I have tried so far is HDR painting which was rather disappointing. I had hoped for better HDR and less painting.

Moving on

Three weeks ago I got a new brain/body for my (camera) lens. I had been using my Nikon D40X with a Nikkor 18-200mm zoom since 25 October 2007, so long enough. The years of use (over 31000 photographs) had baked some dust onto the camera's sensor and these showed up as smudge spots in blue-sky backdrops — very annoying! I suppose this is fixable by a proper in-shop cleaning but the logistics daunted me. So (instead) I purchased a Nikon D5300. It was my intention to take a whole bunch of photos before evaluating the results — but my iMac's two-year-old hard drive died a week into the picture-taking and it took a whole week to replace it. By then, I had decided to upgrade my lens as well. The Nikkor 18-300mm zoom was a near-identical replacement with a small zoom improvement and — more importantly — a lock that prevented the zoom from unfurling under its own weight, something with which I had been struggling all these years.

Where the woodbine twineth

My much-photographed island in the Humber, taken on September 20. Usually I'm high up the bank (behind me) looking down but I wanted to see how close I could get to the egret on the rocks (at bottom right). The woodbine in my title refers to the prominent (red) Virginia creeper (Parthenocissus quinquefolia) climbing up a tree.

What's on first?

A sequence that I'm currently attempting to extend: _, 91, 3854, 178086, 15469622, ... I've purposely left out the surprising first term.

Update: It is now a sequence in the OEIS.

3329 is a prime balanced factorial

It may appear that the following Mathematica execution took just over an hour. In fact, it has taken me many weeks to be able to write it.

PrimeQ[3329!-7607]

True

DateString[]
NextPrime[3329!-7607]
DateString[]

Thu 14 Aug 2014 08:03:36

15015905410377990189612669728734416983623305863211194173984549864118912388614580813647063124137997910538082896578696443428869585291166176049551725545323619613247036490118670691372109371002766714784994640066135388162033055349621680311652435317541931644391114409952234156263908956775470653899105140924947400093330839839804546566886098947871580704800358703875896032243051062750594829590002645279915900517541293263702329765087215432815094241693060954235337839299957090323969065518806385266477265191459751472884324532373143637717234024009643597958281320271908345806890456895205227477366523110847604195929204092016300135023471997451803585110949233566181466735581417716495791896633922227547993640135431551957261121432488358580658859446738751102266955127145501461655564274215743921329774164182560773883535928923859928623997161281386873450899829022827207791609975077260505038324222182602036918649148183767166658704803906524987613581466322117907848171936943276831821376964038916006583422499204626298527081536155227098013942804257772011873945707081139954946428432796314243287864539867192889002088474220653113939262767909977843118614065596100525158318722437297244808511563252129894837185130204981906967470241971751053802841530211141663068662899843720213079867239111965654743779578216120280795733607532383934869566829618551012793251273877554076724502727685976158947950141105949105998628073350927074843754946373489924711282077736360892286369244279778940154812080055330072889475133632730945251644798926678264994198497000992008367717165132808168934785538022596478905028659057420416930567628708103579201145411632305879571078004102460653446380420754587421340701976673333458052068783677292030907456781166992041033853227604884045593985070171656690841917913386767434663058479118452860681284187827048282376831904398605646930051779380911926083051243238455870789275047464245298483271273587903898933743766245002439773172881419876698965849052031018467593963443053676700228214015169134222272062806393035636579926552904633221081761721750145823969767215320529290268701195654553582118809223232013064682937219611566989489148341352831033573418489718988963993993805180906729957424383324212800307148720599750226879494934765856972359660969425261751184149927233027360180134302879130697258875023207844329486751133716567025046003583124738472466899864638004330782854059337089289407398001945157970833338691320373787690787404398358801973738667634465389535587594564920681405752736888311729288272246605928966188557205785691991582812355266808177731327720417562922628147355990730291178024360246732978013628772377136439127156873731248096049600312044261976700965111637819443213683959071375872377007385562407517901374496065791398470908123128307172951685673964210294957748278324201913124374729962664204986302046111534390117942176782243884596262173616164570288711883087828143613585969830142617917940280965919462026407378592510796009217969597008566076229766572343218297868592522669735483527148243498594849361994602293289198078993889348199637296250334805245579058462482111308107426699517465259358792026716200140417484585776804584548583314915343968927754960036162463616241901324430769887203952214802182627959756314445982649524058545311882983999817506462539579403054799862166728589699965459595870991990477454573070360170460041592709595638178559759526185045600203679707267061923408643032494471030635862149983001990503501730690328336124960551908452915041292337581166837138829730178563626402194012313680409767407365529282594987058251861780629567748915228518807383123318556374750937396657975083779591327967943714152154031708084505572098305967040376312526574836658064336135383885522606380468729894197273977828913752002803865934866466084285902953586411476778976985089899167264960569336939422398346244930852202019983334892586386495084115977657343620760816749779886343559438100565067398263630336285623236852512608584848800379595081341663186071192321151990086732363604874257861770988096320978938181949641606264428409380443332636396632465601760606345384780029194243703020573863877365494050356895128710868090837593220445107522893493893198071574776656427848807860183960389256911014990675661921782737633755328058787385898107233714759984556019501248961899288732375240085754014381488919202762095781925676758015829258479348139284759396510731176651509365584251387608639622431879559280845192290535282131606838274408367717288414369423342336700514311757285373699847184742654453078953299536663552867918172295641752756713068488207022447052731729114606154499330438421861227323385195220807906861290851518445313496084054556664268969785557013526231921070169876121224039336072475628070498166383135566735553162710572353048006929601597922923678245925639836253997969994846638653526816282355145653404011655683832008197770593435993335541862133880574772037525994988372621711297467288289070241202104972328279553278766035547237128167481050565685242649901692721364948881010613778779764033462123023908305073504835838647614690148099964153342227610659482874003358958411030166863606333262359230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Thu 14 Aug 2014 09:12:03

A beautiful picture

I have suspected for years that the television picture supplied by the evil Canadian Rogers cable empire was sub-optimal at my Toronto house, owing (I thought) to a less-than-perfect infrastructure situation somewhere upstream. It all came to a head three years ago, as this lengthy complaint to an online Rogers employee attempts to explain. A senior technician did subsequently come to the house and declared my television — not their cabling — to be the problem. And alas, that television did indeed suffer from a defect (unbeknownst to me, as the set was borrowed from my daughter) that prevented proper reception on some channels. Rogers charged me $50 for the technician having to come to my house for no good reason, as it were. In light of the defect, I could not press the issue of sub-optimal reception.

Fast-forward to yesterday morning. I had lost my Teksavvy internet signal (which runs on Rogers' cable) and my television picture was somewhat more degraded than usual (so I knew it was an issue with the cable, not the modem). Rogers demands from Teksavvy a full accounting of the situation before they will issue a ticket for a technician to be sent and so I spent a full hour being interviewed by a Teksavvy employee, gathering modem and network details. Subsequently, Catherine found out from our immediate neighbours that they had lost their Rogers television signal altogether.

Early this morning a couple of Rogers subcontractor vehicles appeared and Catherine learned from their drivers that there were additional houses on the street without a Rogers signal. Unable to fix the problem on their own, they issued a Mayday to Rogers. The Rogers truck (pictured) appeared before ten. By noon I had my internet back — although it was off-again/on-again for several hours as they continued (I guess) to troubleshoot the line. Finally, they were done. I had my internet and when I turned on the television — I had a pristine picture the likes of which I had never seen. So yes, I did have a sub-optimal feed all these years!

Cousins

I thought Kaya and Bodie were step-siblings. Actually, they are cousins. Kaya came for a visit just now. You can see that we've trimmed Bodie's ears and tail.

Bodie

We have adopted a genetically modified wolf. The horror! It's been six-and-a-half years since we lost Micky, a wheaten terrier. Late June we took in (for a week) our friends' miniature poodle, Kaya, and I guess it rekindled in Catherine some dog-ownership benefits. Six-month-old Bodie (rhymes with Jodie) and Kaya are cousins.

Cracks in the crystal

In the late 1980s, I consumed reams of paper printing out (using my HP 75C) evolutions of elementary cellular automaton rule #193 (which in binary is 11000001, suggesting that the central cells of 111, 110, and 000 remain or become 1 in the next generation). This is the mirrored complement of the more famous rule #110. In rule #193, the ones form right "triangles" with a jagged hypotenuse facing southwest. Back then, in order to conserve ink I suppressed the printing of triangles of size-3 which (in evolutions of random starts) soon predominate. Looking at the time history of an evolution, the staggered size-3 triangles form a sort of "crystal" based on the stable configuration of the following cell (left and right are joined, time moves down):

00000100110111

01110000010011

00110111000001

00010011011100

11000001001101

11011100000100

01001101110000

Yesterday I spent some time programming this old plaything in Mathematica. Here is how a larger configuration of crystal cells appears in my implementation:

The lighter-blue ones of the size-3 triangles are meant to blend in with the darker-blue zeros, providing contrast for the other-sized triangles (in orange) of a typical evolution, out of which I have cropped this small detail:

The orange "particles" are recognized as defects, or cracks, in the crystal. They move left or right or just stand still. Colliding particles obviously conserve the sum of the defect offset numbers that individual particles may be said to possess, as they interact and regroup, or occasionally disappear. You can surmise from the (final) four particles at the very bottom of my example that the size of my space is a multiple of 14, the necessary space-size of a left-right-joined perfect crystal. (The right-moving particles will crash into the left-moving particles and disappear.)

A sequence of reasons

Eric Angelini recently posted to the Sequence Fanatics discussion list a very nice 1, 2, 3, 5, 11, 12, 4, 8, 7, ... which ended up as A243357: The lexicographically earliest sequence (not reusing any terms) with the property that if a vertical line is drawn between any pair of adjacent digits (commas and spaces excluded), the number Z formed by the digits to the left of the line is divisible by the final digit of Z. So, a line not just between the terms 3 and 5 (say), or the term 5 and the 1 of the following 11, but also between the two ones of that 11. To wit:

1/1 = 1
12/2 = 6
123/3 = 41
1235/5 = 247
12351/1 = 12351, 123511/1 = 123511
1235111/1 = 1235111, 12351112/2 = 6175556
123511124/4 = 30877781
1235111248/8 = 154388906
12351112487/7 = 1764444641
etc.

Not every number will appear in A243357. Nonnegative integers that are not in A243357 ended up as A244033:

0, 10, 14, 18, 20, 30, 34, 38, 40, 50, 54, 58, 60, 70, 74, 78, 80, 90, 94, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 114, 118, 120, 130, 134, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 154, 158, 160, 170, 174, 178, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 194, 198, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 214, 218, 220, 228, 230, 234, 238, 240, 250, 254, 258, 260, 268, 270, 274, 278, 280, 290, 294, 298, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 313, 314, 316, 318, 319, 320, 323, 326, 329, 330, 334, 338, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 353, 354, 356, 358, 359, 360, 370, 373, 374, 376, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 394, 398, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 414, 418, 420, 428, 430, 434, 438, 440, 450, 454, 458, 460, 468, 470, 474, 478, 480, 490, 494, 498, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 514, 518, 520, 530, 534, 538, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 554, 558, 560, 570, 574, 578, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 594, 598, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 613, 614, 616, 618, 619, 620, 623, 626, 628, 629, 630, 634, 638, 640, 643, 646, 649, 650, 653, 654, 656, 658, 659, 660, 668, 670, 673, 674, 676, 678, 679, 680, 683, 686, 689, 690, 694, 698, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 714, 717, 718, 720, 727, 730, 734, 737, 738, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 754, 757, 758, 760, 767, 770, 774, 778, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 794, 797, 798, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 814, 818, 820, 828, 830, 834, 838, 840, 850, 854, 858, 860, 868, 870, 874, 878, 880, 890, 894, 898, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 913, 914, 916, 918, 919, 920, 923, 926, 929, 930, 934, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 953, 954, 956, 958, 959, 960, 969, 970, 973, 974, 976, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 994, 998, 1000, ...

Most obviously, any number containing a zero digit will have been excluded because division by zero is a no-no. Let's call this reason #1. We can exclude from A244033 those numbers that fail because of reason #1. What remains ended up as A244034:

14, 18, 34, 38, 54, 58, 74, 78, 94, 98, 114, 118, 134, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149, 154, 158, 174, 178, 181, 182, 183, 184, 185, 186, 187, 188, 189, 194, 198, 214, 218, 228, 234, 238, 254, 258, 268, 274, 278, 294, 298, 313, 314, 316, 318, 319, 323, 326, 329, 334, 338, 341, 342, 343, 344, 345, 346, 347, 348, 349, 353, 354, 356, 358, 359, 373, 374, 376, 378, 379, 381, 382, 383, 384, 385, 386, 387, 388, 389, 394, 398, 414, 418, 428, 434, 438, 454, 458, 468, 474, 478, 494, 498, 514, 518, 534, 538, 541, 542, 543, 544, 545, 546, 547, 548, 549, 554, 558, 574, 578, 581, 582, 583, 584, 585, 586, 587, 588, 589, 594, 598, 613, 614, 616, 618, 619, 623, 626, 628, 629, 634, 638, 643, 646, 649, 653, 654, 656, 658, 659, 668, 673, 674, 676, 678, 679, 683, 686, 689, 694, 698, 714, 717, 718, 727, 734, 737, 738, 741, 742, 743, 744, 745, 746, 747, 748, 749, 754, 757, 758, 767, 774, 778, 781, 782, 783, 784, 785, 786, 787, 788, 789, 794, 797, 798, 814, 818, 828, 834, 838, 854, 858, 868, 874, 878, 894, 898, 913, 914, 916, 918, 919, 923, 926, 929, 934, 938, 939, 941, 942, 943, 944, 945, 946, 947, 948, 949, 953, 954, 956, 958, 959, 969, 973, 974, 976, 978, 979, 981, 982, 983, 984, 985, 986, 987, 988, 989, 994, 998, 1114, ...

Why do these numbers fail to be included in A243357? Examining the initial ten terms, one realizes soon enough that multiples of 4 ending in 4 and multiples of 8 ending in 8 will always have an even digit preceding the 4 or 8. Therefore 14, 18, 34, 38, 54, 58, 74, 78, 94, 98, or any larger number containing these as a substring, will have been excluded for that reason. Let's call this reason #2. We can exclude from A244034 those numbers that fail because of reason #2. What remains is:

228, 268, 313, 316, 319, 323, 326, 329, 353, 356, 359, 373, 376, 379, 428, 468, 613, 616, 619, 623, 626, 628, 629, 643, 646, 649, 653, 656, 659, 668, 673, 676, 679, 683, 686, 689, 717, 727, 737, 757, 767, 797, 828, 868, 913, 916, 919, 923, 926, 929, 939, 953, 956, 959, 969, 973, 976, 979, 1228, ...

Why do these numbers fail to be included in A243357? Not because of reason #1. Not because of reason #2. You can see where I'm going with this. Without further ado, I present reasons to exclude from A243357 certain numbers:

#1: numbers containing one or more of the single-digit 0.

#2: numbers containing as a substring any one or more of the 2-digit strings 14, 18, 34, 38, 54, 58, 74, 78, 94, 98.

#3: numbers containing as a substring any one or more of the 3-digit strings 228, 268, 313, 316, 319, 323, 326, 329, 353, 356, 359, 373, 376, 379, 428, 468, 613, 616, 619, 623, 626, 628, 629, 643, 646, 649, 653, 656, 659, 668, 673, 676, 679, 683, 686, 689, 717, 727, 737, 757, 767, 797, 828, 868, 913, 916, 919, 923, 926, 929, 939, 953, 956, 959, 969, 973, 976, 979.

#4: numbers containing as a substring any one or more of the 4-digit strings 3113, 3116, 3119, 3173, 3176, 3179, 3223, 3226, 3229, 3253, 3256, 3259, 3283, 3286, 3289, 3523, 3526, 3529, 3553, 3556, 3559, 3713, 3716, 3719, 3773, 3776, 3779, 6113, 6116, 6119, 6173, 6176, 6179, 6223, 6226, 6229, 6253, 6256, 6259, 6413, 6416, 6419, 6443, 6446, 6449, 6473, 6476, 6479, 6523, 6526, 6529, 6553, 6556, 6559, 6713, 6716, 6719, 6773, 6776, 6779, 6823, 6826, 6829, 6853, 6856, 6859, 6883, 6886, 6889, 7117, 7127, 7137, 7157, 7167, 7197, 7227, 7237, 7247, 7257, 7267, 7297, 7317, 7327, 7337, 7367, 7397, 7517, 7527, 7537, 7557, 7597, 7617, 7627, 7647, 7657, 7667, 7687, 7697, 7927, 7937, 7957, 7967, 7997, 9113, 9116, 9119, 9129, 9159, 9173, 9176, 9179, 9219, 9223, 9226, 9229, 9249, 9253, 9256, 9259, 9283, 9286, 9289, 9339, 9519, 9523, 9526, 9529, 9553, 9556, 9559, 9579, 9669, 9713, 9716, 9719, 9759, 9773, 9776, 9779.

#5: numbers containing as a substring any one or more of the 5-digit strings 31123, 31126, 31129, 31153, 31156, 31159, 31213, 31216, 31219, 31223, 31226, 31229, 31243, 31246, 31249, 31253, 31256, 31259, 31273, 31276, 31279, 31283, 31286, 31289, 31513, 31516, 31519, 31523, 31526, 31529, 31553, 31556, 31559, 31573, 31576, 31579, 31723, 31726, 31729, 31753, 31756, 31759, 32113, 32116, 32119, 32123, 32126, 32129, 32153, 32156, 32159, 32173, 32176, 32179, 32213, 32216, 32219, 32243, 32246, 32249, 32273, 32276, 32279, 32413, 32416, 32419, 32423, 32426, 32429, 32443, 32446, 32449, 32453, 32456, 32459, 32473, 32476, 32479, 32483, 32486, 32489, 32513, 32516, 32519, 32573, 32576, 32579, 32713, 32716, 32719, 32723, 32726, 32729, 32753, 32756, 32759, 32773, 32776, 32779, 32813, 32816, 32819, 32843, 32846, 32849, 32873, 32876, 32879, 35113, 35116, 35119, 35123, 35126, 35129, 35153, 35156, 35159, 35173, 35176, 35179, 35213, 35216, 35219, 35243, 35246, 35249, 35273, 35276, 35279, 35513, 35516, 35519, 35573, 35576, 35579, 35713, 35716, 35719, 35723, 35726, 35729, 35753, 35756, 35759, 35773, 35776, 35779, 37123, 37126, 37129, 37153, 37156, 37159, 37213, 37216, 37219, 37223, 37226, 37229, 37243, 37246, 37249, 37253, 37256, 37259, 37283, 37286, 37289, 37513, 37516, 37519, 37523, 37526, 37529, 37553, 37556, 37559, 37723, 37726, 37729, 37753, 37756, 37759, 61123, 61126, 61129, 61153, 61156, 61159, 61213, 61216, 61219, 61223, 61226, 61229, 61243, 61246, 61249, 61253, 61256, 61259, 61273, 61276, 61279, 61283, 61286, 61289, 61513, 61516, 61519, 61523, 61526, 61529, 61553, 61556, 61559, 61573, 61576, 61579, 61723, 61726, 61729, 61753, 61756, 61759, 62113, 62116, 62119, 62123, 62126, 62129, 62153, 62156, 62159, 62173, 62176, 62179, 62213, 62216, 62219, 62243, 62246, 62249, 62273, 62276, 62279, 62413, 62416, 62419, 62423, 62426, 62429, 62443, 62446, 62449, 62453, 62456, 62459, 62473, 62476, 62479, 62483, 62486, 62489, 62513, 62516, 62519, 62573, 62576, 62579, 62713, 62716, 62719, 62723, 62726, 62729, 62753, 62756, 62759, 62773, 62776, 62779, 64123, 64126, 64129, 64153, 64156, 64159, 64213, 64216, 64219, 64223, 64226, 64229, 64243, 64246, 64249, 64253, 64256, 64259, 64273, 64276, 64279, 64423, 64426, 64429, 64453, 64456, 64459, 64483, 64486, 64489, 64513, 64516, 64519, 64523, 64526, 64529, 64553, 64556, 64559, 64573, 64576, 64579, 64723, 64726, 64729, 64753, 64756, 64759, 64813, 64816, 64819, 64823, 64826, 64829, 64843, 64846, 64849, 64853, 64856, 64859, 64873, 64876, 64879, 64883, 64886, 64889, 65113, 65116, 65119, 65123, 65126, 65129, 65153, 65156, 65159, 65173, 65176, 65179, 65213, 65216, 65219, 65243, 65246, 65249, 65273, 65276, 65279, 65513, 65516, 65519, 65573, 65576, 65579, 65713, 65716, 65719, 65723, 65726, 65729, 65753, 65756, 65759, 65773, 65776, 65779, 67123, 67126, 67129, 67153, 67156, 67159, 67213, 67216, 67219, 67223, 67226, 67229, 67243, 67246, 67249, 67253, 67256, 67259, 67283, 67286, 67289, 67513, 67516, 67519, 67523, 67526, 67529, 67553, 67556, 67559, 67723, 67726, 67729, 67753, 67756, 67759, 68113, 68116, 68119, 68123, 68126, 68129, 68153, 68156, 68159, 68173, 68176, 68179, 68213, 68216, 68219, 68243, 68246, 68249, 68273, 68276, 68279, 68413, 68416, 68419, 68423, 68426, 68429, 68443, 68446, 68449, 68453, 68456, 68459, 68473, 68476, 68479, 68483, 68486, 68489, 68513, 68516, 68519, 68573, 68576, 68579, 68713, 68716, 68719, 68723, 68726, 68729, 68753, 68756, 68759, 68773, 68776, 68779, 68813, 68816, 68819, 68843, 68846, 68849, 68873, 68876, 68879, 71117, 71137, 71157, 71167, 71217, 71227, 71237, 71247, 71257, 71287, 71297, 71317, 71327, 71357, 71367, 71397, 71517, 71527, 71537, 71557, 71567, 71597, 71627, 71637, 71647, 71657, 71667, 71697, 71917, 71927, 71937, 71957, 71997, 72117, 72127, 72137, 72157, 72167, 72197, 72217, 72227, 72237, 72257, 72267, 72297, 72327, 72337, 72357, 72367, 72397, 72417, 72427, 72437, 72447, 72467, 72487, 72497, 72517, 72537, 72557, 72567, 72617, 72627, 72637, 72647, 72657, 72697, 72817, 72827, 72837, 72847, 72857, 72867, 72887, 72897, 72917, 72927, 72937, 72957, 72967, 72997, 73117, 73127, 73217, 73247, 73257, 73287, 73317, 73327, 73337, 73357, 73397, 73517, 73527, 73557, 73617, 73627, 73637, 73657, 73667, 73687, 73697, 73917, 73937, 73957, 73967, 75127, 75137, 75157, 75167, 75197, 75217, 75227, 75237, 75247, 75267, 75287, 75297, 75317, 75337, 75357, 75367, 75517, 75527, 75557, 75567, 75597, 75617, 75627, 75637, 75647, 75657, 75667, 75687, 75697, 75917, 75927, 75937, 75967, 75997, 76117, 76127, 76157, 76217, 76227, 76247, 76257, 76317, 76327, 76337, 76357, 76367, 76397, 76417, 76427, 76457, 76487, 76527, 76557, 76617, 76627, 76637, 76647, 76667, 76697, 76817, 76827, 76847, 76857, 76887, 76917, 76927, 76957, 76967, 76997, 79117, 79127, 79157, 79217, 79227, 79257, 79287, 79327, 79337, 79357, 79367, 79517, 79557, 79617, 79627, 79637, 79647, 79657, 79687, 79917, 79927, 79937, 79957, 79967, 79997, 91119, 91123, 91126, 91129, 91153, 91156, 91159, 91213, 91216, 91219, 91223, 91226, 91229, 91239, 91243, 91246, 91249, 91253, 91256, 91259, 91273, 91276, 91279, 91283, 91286, 91289, 91513, 91516, 91519, 91523, 91526, 91529, 91553, 91556, 91559, 91569, 91573, 91576, 91579, 91723, 91726, 91729, 91753, 91756, 91759, 91779, 92113, 92116, 92119, 92123, 92126, 92129, 92139, 92153, 92156, 92159, 92173, 92176, 92179, 92213, 92216, 92219, 92229, 92243, 92246, 92249, 92273, 92276, 92279, 92413, 92416, 92419, 92423, 92426, 92429, 92443, 92446, 92449, 92453, 92456, 92459, 92469, 92473, 92476, 92479, 92483, 92486, 92489, 92513, 92516, 92519, 92559, 92573, 92576, 92579, 92713, 92716, 92719, 92723, 92726, 92729, 92739, 92753, 92756, 92759, 92769, 92773, 92776, 92779, 92813, 92816, 92819, 92829, 92843, 92846, 92849, 92859, 92873, 92876, 92879, 93129, 93219, 93279, 93369, 93579, 93639, 93669, 93729, 93759, 95113, 95116, 95119, 95123, 95126, 95129, 95153, 95156, 95159, 95169, 95173, 95176, 95179, 95213, 95216, 95219, 95243, 95246, 95249, 95259, 95273, 95276, 95279, 95289, 95513, 95516, 95519, 95529, 95559, 95573, 95576, 95579, 95713, 95716, 95719, 95723, 95726, 95729, 95739, 95753, 95756, 95759, 95773, 95776, 95779, 96159, 96249, 96279, 96339, 96369, 96429, 96459, 96519, 96639, 96729, 96819, 96879, 97123, 97126, 97129, 97153, 97156, 97159, 97213, 97216, 97219, 97223, 97226, 97229, 97239, 97243, 97246, 97249, 97253, 97256, 97259, 97269, 97283, 97286, 97289, 97513, 97516, 97519, 97523, 97526, 97529, 97539, 97553, 97556, 97559, 97719, 97723, 97726, 97729, 97753, 97756, 97759, 97779.

#6: numbers containing as a substring any one or more of the 6-digit strings 311123, 311126, 311129, 311153, 311156, 311159, 311213, 311216, 311219, 311243, 311246, 311249, 311273, 311276, 311279, 311513, 311516, 311519, 311573, 311576, 311579, 311723, 311726, 311729, 311753, 311756, 311759, 312113, 312116, 312119, 312173, 312176, 312179, 312413, 312416, 312419, 312443, 312446, 312449, 312473, 312476, 312479, 312713, 312716, 312719, 312773, 312776, 312779, 315113, 315116, 315119, 315173, 315176, 315179, 315713, 315716, 315719, 315773, 315776, 315779, 317123, 317126, 317129, 317153, 317156, 317159, 317213, 317216, 317219, 317243, 317246, 317249, 317513, 317516, 317519, 317723, 317726, 317729, 317753, 317756, 317759, 321113, 321116, 321119, 321173, 321176, 321179, 321713, 321716, 321719, 321773, 321776, 321779, 322223, 322226, 322229, 322253, 322256, 322259, 322523, 322526, 322529, 322553, 322556, 322559, 324113, 324116, 324119, 324173, 324176, 324179, 324413, 324416, 324419, 324443, 324446, 324449, 324473, 324476, 324479, 324713, 324716, 324719, 324773, 324776, 324779, 325223, 325226, 325229, 325253, 325256, 325259, 325283, 325286, 325289, 325523, 325526, 325529, 325553, 325556, 325559, 327113, 327116, 327119, 327713, 327716, 327719, 327773, 327776, 327779, 328223, 328226, 328229, 328253, 328256, 328259, 328523, 328526, 328529, 328553, 328556, 328559, 328823, 328826, 328829, 328853, 328856, 328859, 328883, 328886, 328889, 351113, 351116, 351119, 351173, 351176, 351179, 351713, 351716, 351719, 351773, 351776, 351779, 352223, 352226, 352229, 352253, 352256, 352259, 352523, 352526, 352529, 352553, 352556, 352559, 352823, 352826, 352829, 352853, 352856, 352859, 352883, 352886, 352889, 355223, 355226, 355229, 355253, 355256, 355259, 355283, 355286, 355289, 355523, 355526, 355529, 355553, 355556, 355559, 357113, 357116, 357119, 357713, 357716, 357719, 357773, 357776, 357779, 371123, 371126, 371129, 371153, 371156, 371159, 371213, 371216, 371219, 371243, 371246, 371249, 371513, 371516, 371519, 372113, 372116, 372119, 372173, 372176, 372179, 372413, 372416, 372419, 372443, 372446, 372449, 375113, 375116, 375119, 377123, 377126, 377129, 377153, 377156, 377159, 377213, 377216, 377219, 377243, 377246, 377249, 377513, 377516, 377519, 377723, 377726, 377729, 377753, 377756, 377759, 611123, 611126, 611129, 611153, 611156, 611159, 611213, 611216, 611219, 611243, 611246, 611249, 611273, 611276, 611279, 611513, 611516, 611519, 611573, 611576, 611579, 611723, 611726, 611729, 611753, 611756, 611759, 612113, 612116, 612119, 612173, 612176, 612179, 612413, 612416, 612419, 612443, 612446, 612449, 612473, 612476, 612479, 612713, 612716, 612719, 612773, 612776, 612779, 615113, 615116, 615119, 615173, 615176, 615179, 615713, 615716, 615719, 615773, 615776, 615779, 617123, 617126, 617129, 617153, 617156, 617159, 617213, 617216, 617219, 617243, 617246, 617249, 617513, 617516, 617519, 617723, 617726, 617729, 617753, 617756, 617759, 621113, 621116, 621119, 621173, 621176, 621179, 621713, 621716, 621719, 621773, 621776, 621779, 622223, 622226, 622229, 622253, 622256, 622259, 622523, 622526, 622529, 622553, 622556, 622559, 624113, 624116, 624119, 624173, 624176, 624179, 624413, 624416, 624419, 624443, 624446, 624449, 624473, 624476, 624479, 624713, 624716, 624719, 624773, 624776, 624779, 625223, 625226, 625229, 625253, 625256, 625259, 625283, 625286, 625289, 625523, 625526, 625529, 625553, 625556, 625559, 627113, 627116, 627119, 627713, 627716, 627719, 627773, 627776, 627779, 641123, 641126, 641129, 641153, 641156, 641159, 641213, 641216, 641219, 641243, 641246, 641249, 641273, 641276, 641279, 641513, 641516, 641519, 641573, 641576, 641579, 641723, 641726, 641729, 641753, 641756, 641759, 642113, 642116, 642119, 642173, 642176, 642179, 642413, 642416, 642419, 642443, 642446, 642449, 642473, 642476, 642479, 642713, 642716, 642719, 642773, 642776, 642779, 644123, 644126, 644129, 644153, 644156, 644159, 644213, 644216, 644219, 644243, 644246, 644249, 644273, 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655283, 655286, 655289, 655523, 655526, 655529, 655553, 655556, 655559, 657113, 657116, 657119, 657713, 657716, 657719, 657773, 657776, 657779, 671123, 671126, 671129, 671153, 671156, 671159, 671213, 671216, 671219, 671243, 671246, 671249, 671513, 671516, 671519, 672113, 672116, 672119, 672173, 672176, 672179, 672413, 672416, 672419, 672443, 672446, 672449, 675113, 675116, 675119, 677123, 677126, 677129, 677153, 677156, 677159, 677213, 677216, 677219, 677243, 677246, 677249, 677513, 677516, 677519, 677723, 677726, 677729, 677753, 677756, 677759, 681113, 681116, 681119, 681173, 681176, 681179, 681713, 681716, 681719, 681773, 681776, 681779, 682223, 682226, 682229, 682253, 682256, 682259, 682523, 682526, 682529, 682553, 682556, 682559, 684113, 684116, 684119, 684173, 684176, 684179, 684413, 684416, 684419, 684443, 684446, 684449, 684473, 684476, 684479, 684713, 684716, 684719, 684773, 684776, 684779, 685223, 685226, 685229, 685253, 685256, 685259, 685283, 685286, 685289, 685523, 685526, 685529, 685553, 685556, 685559, 687113, 687116, 687119, 687713, 687716, 687719, 687773, 687776, 687779, 688223, 688226, 688229, 688253, 688256, 688259, 688523, 688526, 688529, 688553, 688556, 688559, 688823, 688826, 688829, 688853, 688856, 688859, 688883, 688886, 688889, 711117, 711157, 711167, 711217, 711227, 711237, 711247, 711257, 711287, 711297, 711317, 711327, 711357, 711367, 711397, 711517, 711527, 711537, 711567, 711597, 711637, 711647, 711657, 711667, 711917, 711927, 711937, 711957, 711997, 712127, 712137, 712157, 712167, 712197, 712217, 712227, 712237, 712267, 712297, 712337, 712357, 712367, 712417, 712427, 712437, 712447, 712487, 712497, 712517, 712557, 712567, 712617, 712627, 712637, 712647, 712657, 712697, 712827, 712837, 712847, 712857, 712867, 712897, 712917, 712927, 712937, 712967, 712997, 713117, 713127, 713217, 713247, 713257, 713287, 713317, 713327, 713337, 713357, 713397, 713527, 713557, 713617, 713627, 713637, 713667, 713687, 713697, 713917, 713957, 713967, 715137, 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722517, 722527, 722557, 722567, 722597, 722627, 722637, 722647, 722657, 722667, 722697, 722917, 722927, 722937, 722997, 723117, 723127, 723157, 723217, 723227, 723257, 723287, 723327, 723337, 723357, 723367, 723397, 723557, 723617, 723627, 723637, 723647, 723687, 723697, 723917, 723927, 723957, 723967, 723997, 724117, 724127, 724137, 724167, 724197, 724237, 724247, 724257, 724267, 724317, 724327, 724337, 724397, 724417, 724447, 724457, 724467, 724487, 724517, 724527, 724537, 724557, 724597, 724617, 724627, 724657, 724667, 724697, 724817, 724827, 724837, 724867, 724887, 724897, 724937, 724957, 724967, 725117, 725157, 725167, 725217, 725227, 725237, 725247, 725257, 725287, 725297, 725317, 725327, 725357, 725367, 725397, 725517, 725527, 725537, 725567, 725597, 725637, 725647, 725657, 725667, 725917, 725927, 725937, 725957, 725997, 726127, 726157, 726217, 726227, 726337, 726357, 726367, 726417, 726427, 726447, 726487, 726517, 726557, 726617, 726627, 726637, 726647, 726657, 726697, 726917, 726927, 726937, 726967, 726997, 728117, 728127, 728157, 728167, 728197, 728227, 728237, 728247, 728257, 728267, 728297, 728317, 728327, 728337, 728367, 728397, 728437, 728447, 728457, 728467, 728517, 728527, 728537, 728597, 728617, 728647, 728657, 728667, 728817, 728827, 728857, 728867, 728887, 728897, 728927, 728937, 728957, 728967, 728997, 729157, 729217, 729227, 729247, 729287, 729317, 729357, 729367, 729517, 729527, 729557, 729627, 729637, 729647, 729657, 729667, 729917, 729927, 729937, 729997, 731117, 731127, 731237, 731247, 731257, 731267, 731517, 731527, 731537, 731557, 731597, 732117, 732157, 732167, 732217, 732227, 732247, 732257, 732427, 732437, 732447, 732457, 732467, 732497, 732517, 732527, 732817, 732847, 732857, 732887, 733127, 733157, 733217, 733227, 733287, 733337, 733357, 733367, 733517, 733557, 733617, 733627, 733637, 733647, 733657, 733687, 733697, 733917, 733927, 733937, 733967, 733997, 735117, 735127, 735157, 735167, 735197, 735227, 735247, 735257, 735517, 735527, 736157, 736217, 736227, 736247, 736317, 736357, 736367, 736417, 736427, 736447, 736457, 736487, 736517, 736527, 736557, 736627, 736637, 736647, 736657, 736667, 736697, 736847, 736857, 736917, 736927, 736937, 736997, 739117, 739157, 739217, 739227, 739247, 739257, 739287, 739317, 739327, 739357, 739367, 739517, 739527, 739637, 739647, 739657, 739667, 739917, 739927, 739937, 739957, 739997, 751117, 751127, 751137, 751157, 751197, 751217, 751227, 751257, 751267, 751287, 751297, 751327, 751337, 751357, 751367, 751397, 751537, 751557, 751567, 751617, 751627, 751637, 751647, 751687, 751697, 751917, 751927, 751957, 751967, 751997, 752117, 752127, 752137, 752167, 752197, 752237, 752247, 752257, 752267, 752317, 752327, 752337, 752397, 752417, 752447, 752457, 752467, 752487, 752517, 752527, 752537, 752557, 752597, 752617, 752627, 752657, 752667, 752697, 752817, 752827, 752837, 752867, 752887, 752897, 752937, 752957, 752967, 753117, 753157, 753217, 753227, 753247, 753257, 753287, 753317, 753327, 753357, 753367, 753397, 753517, 753527, 753637, 753647, 753657, 753667, 753917, 753927, 753937, 753957, 753997, 755117, 755127, 755137, 755197, 755217, 755247, 755257, 755267, 755287, 755317, 755327, 755337, 755357, 755397, 755527, 755537, 755557, 755567, 755597, 755617, 755627, 755637, 755667, 755687, 755697, 755917, 755957, 755967, 756117, 756127, 756157, 756227, 756247, 756257, 756317, 756327, 756337, 756367, 756397, 756447, 756457, 756517, 756527, 756617, 756647, 756657, 756667, 756817, 756827, 756857, 756887, 756927, 756937, 756957, 756967, 756997, 759117, 759127, 759247, 759257, 759317, 759327, 759337, 759517, 759527, 759557, 759617, 759627, 759657, 759667, 759687, 759937, 759957, 759967, 761127, 761157, 761217, 761227, 761237, 761267, 761287, 761297, 761517, 761557, 761567, 762117, 762127, 762137, 762197, 762217, 762247, 762257, 762417, 762427, 762457, 762467, 762487, 762497, 762527, 762557, 763117, 763127, 763157, 763227, 763247, 763257, 763317, 763327, 763337, 763367, 763397, 763517, 763527, 763617, 763647, 763657, 763667, 763687, 763927, 763937, 763957, 763967, 763997, 764157, 764217, 764227, 764237, 764247, 764297, 764417, 764427, 764447, 764457, 764487, 764517, 764527, 764557, 764567, 764597, 764837, 764847, 764857, 764867, 765117, 765127, 765137, 765157, 765197, 765217, 765227, 765257, 765287, 765557, 766117, 766127, 766247, 766257, 766317, 766327, 766337, 766397, 766417, 766447, 766457, 766487, 766517, 766527, 766557, 766617, 766627, 766657, 766667, 766697, 766937, 766957, 766967, 768127, 768137, 768157, 768167, 768197, 768217, 768227, 768417, 768427, 768437, 768447, 768487, 768497, 768517, 768557, 768827, 768847, 768857, 769117, 769127, 769217, 769247, 769257, 769287, 769317, 769327, 769337, 769357, 769527, 769557, 769617, 769627, 769637, 769667, 769687, 769917, 769957, 769967, 791117, 791127, 791157, 791227, 791237, 791247, 791257, 791267, 791517, 791527, 791537, 792137, 792157, 792167, 792217, 792227, 792247, 792417, 792427, 792437, 792447, 792457, 792487, 792517, 792527, 792557, 792847, 792857, 793117, 793127, 793157, 793217, 793227, 793257, 793287, 793327, 793337, 793357, 793367, 793557, 793617, 793627, 793637, 793647, 793687, 793697, 795117, 795157, 795167, 795217, 795227, 795247, 795257, 795287, 795517, 795527, 796127, 796157, 796217, 796227, 796337, 796357, 796367, 796417, 796427, 796447, 796487, 796517, 796557, 796617, 796627, 796637, 796647, 796657, 796827, 796847, 796857, 799157, 799217, 799227, 799247, 799287, 799317, 799357, 799367, 799517, 799527, 799557, 799627, 799637, 799647, 799657, 799667, 799917, 799927, 799937, 799997, 911123, 911126, 911129, 911139, 911153, 911156, 911159, 911179, 911213, 911216, 911219, 911229, 911243, 911246, 911249, 911273, 911276, 911279, 911289, 911513, 911516, 911519, 911559, 911573, 911576, 911579, 911719, 911723, 911726, 911729, 911739, 911753, 911756, 911759, 911769, 911779, 912113, 912116, 912119, 912129, 912173, 912176, 912179, 912219, 912229, 912259, 912279, 912369, 912413, 912416, 912419, 912443, 912446, 912449, 912459, 912473, 912476, 912479, 912489, 912529, 912579, 912639, 912669, 912713, 912716, 912719, 912729, 912759, 912773, 912776, 912779, 912819, 912849, 912859, 912889, 915113, 915116, 915119, 915159, 915173, 915176, 915179, 915229, 915249, 915279, 915289, 915339, 915369, 915519, 915559, 915639, 915713, 915716, 915719, 915729, 915773, 915776, 915779, 917119, 917123, 917126, 917129, 917139, 917153, 917156, 917159, 917169, 917213, 917216, 917219, 917229, 917243, 917246, 917249, 917259, 917513, 917516, 917519, 917529, 917719, 917723, 917726, 917729, 917753, 917756, 917759, 917769, 921113, 921116, 921119, 921129, 921173, 921176, 921179, 921219, 921229, 921259, 921279, 921369, 921529, 921579, 921639, 921669, 921713, 921716, 921719, 921729, 921759, 921773, 921776, 921779, 922119, 922129, 922159, 922179, 922219, 922223, 922226, 922229, 922249, 922253, 922256, 922259, 922269, 922429, 922449, 922479, 922489, 922519, 922523, 922526, 922529, 922539, 922553, 922556, 922559, 922569, 922579, 922719, 922759, 924113, 924116, 924119, 924159, 924173, 924176, 924179, 924229, 924249, 924279, 924339, 924369, 924413, 924416, 924419, 924429, 924443, 924446, 924449, 924459, 924473, 924476, 924479, 924519, 924559, 924639, 924713, 924716, 924719, 924729, 924773, 924776, 924779, 924819, 924829, 924859, 924879, 925129, 925179, 925219, 925223, 925226, 925229, 925239, 925253, 925256, 925259, 925269, 925279, 925283, 925286, 925289, 925523, 925526, 925529, 925539, 925553, 925556, 925559, 925579, 925719, 925729, 925759, 925779, 927113, 927116, 927119, 927129, 927159, 927219, 927249, 927259, 927289, 927339, 927519, 927529, 927559, 927669, 927713, 927716, 927719, 927759, 927773, 927776, 927779, 928119, 928159, 928223, 928226, 928229, 928239, 928249, 928253, 928256, 928259, 928279, 928419, 928429, 928459, 928479, 928519, 928523, 928526, 928529, 928553, 928556, 928559, 928569, 928729, 928779, 928819, 928823, 928826, 928829, 928839, 928853, 928856, 928859, 928869, 928879, 928883, 928886, 928889, 931119, 931179, 931269, 931539, 931569, 931719, 932169, 932259, 932439, 932469, 932529, 932559, 932739, 932829, 932889, 933159, 933249, 933279, 933339, 933369, 933519, 933639, 933729, 935139, 935169, 935229, 935259, 935529, 935769, 936129, 936159, 936219, 936249, 936339, 936429, 936489, 936519, 936579, 936669, 936759, 936849, 936879, 937119, 937239, 937569, 937779, 951113, 951116, 951119, 951159, 951173, 951176, 951179, 951229, 951249, 951279, 951289, 951339, 951369, 951519, 951559, 951639, 951713, 951716, 951719, 951729, 951773, 951776, 951779, 952129, 952179, 952219, 952223, 952226, 952229, 952239, 952253, 952256, 952259, 952269, 952279, 952419, 952449, 952459, 952489, 952523, 952526, 952529, 952539, 952553, 952556, 952559, 952579, 952719, 952729, 952759, 952779, 952819, 952823, 952826, 952829, 952849, 952853, 952856, 952859, 952869, 952883, 952886, 952889, 955119, 955159, 955223, 955226, 955229, 955239, 955249, 955253, 955256, 955259, 955279, 955283, 955286, 955289, 955519, 955523, 955526, 955529, 955553, 955556, 955559, 955569, 955729, 955779, 957113, 957116, 957119, 957129, 957219, 957229, 957259, 957369, 957529, 957639, 957669, 957713, 957716, 957719, 957729, 957759, 957773, 957776, 957779, 961179, 961239, 961269, 961539, 961719, 961779, 962139, 962169, 962229, 962259, 962439, 962529, 962769, 963129, 963159, 963219, 963249, 963339, 963519, 963579, 963669, 963759, 964119, 964239, 964419, 964479, 964569, 964779, 964839, 964869, 965139, 965229, 965289, 965559, 965739, 965769, 966129, 966219, 966279, 966369, 966459, 966489, 966579, 966639, 966669, 966729, 966759, 967119, 967269, 967539, 967569, 967719, 968169, 968259, 968439, 968469, 968529, 968559, 968739, 968829, 968889, 971119, 971123, 971126, 971129, 971139, 971153, 971156, 971159, 971169, 971213, 971216, 971219, 971229, 971243, 971246, 971249, 971259, 971513, 971516, 971519, 971529, 972113, 972116, 972119, 972129, 972159, 972173, 972176, 972179, 972219, 972249, 972259, 972339, 972413, 972416, 972419, 972429, 972443, 972446, 972449, 972489, 972519, 972529, 972559, 972669, 972829, 972849, 972879, 972889, 975113, 975116, 975119, 975129, 975219, 975229, 975259, 975369, 975529, 975639, 975669, 977119, 977123, 977126, 977129, 977153, 977156, 977159, 977169, 977213, 977216, 977219, 977243, 977246, 977249, 977259, 977289, 977513, 977516, 977519, 977529, 977559, 977723, 977726, 977729, 977739, 977753, 977756, 977759, 977779.

And so on. Obviously, numbers may fail to be included in A243357 for more than one reason. For example, 140 fails because of reason #1 and reason #2. The numbers in each reason are a kind of puzzle asking you to determine why that number fails. Once the eight integers ending with 8 in reason #3 are dispensed with, it appears that every other number involves some sort of division complication of integers beginning and ending with 3, 6, or 9, or beginning and ending with 7.

Mac Pro update

I'm coming up to five months usage of my Mac Pro. I long ago disconnected it from the living room TV and placed it on the high plateau at the back of my rolltop desk, hidden from view for the most part by my iMac. I am happily using wireless screen sharing to interact with it but I did order a ThunderBolt cable so that I might do so directly via Target Display Mode.

The screen grab shows what I am doing on my Pro. On the top left is Mathematica calculating an extension to A066364, something I have been working on for quite a while. I'm near 2000 terms and I have another two weeks to go before I reach 10^12. Below Mathematica is my Activity Monitor showing the top active processes. My Pro has only six cores but through some sort of doubling magic the sum of all processes can approach 1200% CPU. At bottom center is my dock, and behind it my Terminal window which I use only to run Dario Alpern's java factorization app. In fact, I have nine of these running (top and right) working on 120-digit composites. The seventh one, just above the Terminal, has found a 42-digit factor (highlighted in blue). Bottom right is a hint of my Finder/desktop.

9^9^9^9

Evaluated of course from the top down: 9^(9^(9^9)) =

21419832947968056113333643734424808301472270728451284887065161959828087496567048470361184472499173685348825764518319411249675059163057939450132383137857257387303899906076223751642158506081537516902249534368182485563436460743204914221083241359509171979501742403737274498190992986234420741963952331470128519454586086183389560459447508519174567154661923513414667736134400991225435672831447817195155379261398749901736062235194235305257317470352535034598833851235160105776671276946104531563857467658976215624043993795779561336003808688926266884283761352441497964782806972761629249342998478005772448909628121104203754931573509482876386281192028953230881806350058010597823777723094132626264578746316256673928426734253433557310495252276792426000946342114034655162413993889346040025558488501379489211584073640000670401588755083736879801550128723434452732373594188602708684863434667860439349440029670400440211337834144962661525484940521999586681805342308223638014652970224685931565568787952219583051583760692665335728780491120909774512599634140270525211434950910562004862991500563860362784147113854804111721510669363426192871540771532588537785733596195270893543172952866225695049165504170200387017043648598016350921587164435363590773303193710778281500541706099414499374130783289007571097319991330040793257406144794022866312593734363512505028018181757699431623730362825141400084201377732055413184733884246186781690131481032352870071901478070878002728132575826640389832532452337525474162592160525288746795727173901735308910110418559632856038429026600648046284160478291787441379326887894347380169206804072022656881349795957890411151700619897854232776881602660007591495541686357630643413099875580703711848095192915749513710023681426562083696413249244304846933560112450385621972240709351464870723879171250706461876757276382612247791670102965366491521477461192657286286686997435699646809578084008568030860483298051587062407328939846546044851729698471329507733283005454052568763236711889000213264716126852718152634154278324290850376993674713864120455120213670722146476721225325078575154901670322007726387082586092730520192259534163815879741736191752594107802548859899652249831169519173588631223437682950230138533178252678097248936287545550539531790153295476611233774538079213115131372514203838588023224526073699784338233113989108244605285009206161082435622947620678974760588393116114349053742518934087148032516574282470515947852012334409693853256811465531586246665952617532044489026105586454007354500639698864881481870834540293047438520744225330668259771869163603116541965700588490445673431619614762936509103240863885315919317042736999706475202464717014802207877539565433704962457775870709201947013460276420454955168669179942404186367520759098483957187057667687422498148163938227735579656358915661264479854579436881770645936131226513142519724463042874418807189979874674266750389040842107384189613419938737245821675358400872657469614579704360701127366825258542791295924213240392280570173797074000109966557177865975907103462288638372720801043868399220499501845883108175247867245503293313884441066362667722424940398617732320900366045529821408386312046037549299194758531800396171364286076210238412059277853224491402153094099140931845421823330466494048710696221318746867639564959692431682505296039969354814380499445300800917574017198570878762399458795171318508568332984880615336673462004387365505518127371012997914806708131522105101294255628149198107105629222041915747683741984328947471716000663006172812933414770723169962371393963681187382384701871358232965345968827185739888932658209796139587407752076843272017801027363325505620993374150100380345978437169022059348662355392750543962418176135982953929193245881871992577351288227199525525672162777037369779993842507258983012507005230908194449478278767115369152101406777750616299533255032976571201848676526916238407266722539339703860141178186954328277654012563050254381444509878281404729668264995808316604156288715005059915700303502946028741087979953976431522765778877235981384740818778166658825968527102178265345054485558256232242739367400360374491226506588072144365872765202068905635109399653350047803247595332564428122409532102816104552855202153189018603947244297559027570339268573460527499371751164149140297165476333046781060914784845729334021263001078166991739266659151865514344968369470546121206451208260149094271086957771896787390624887863768384968986158459649965518375616802530458614364453185879824811106315439747116815133445128326680457044152404890438589029224177709276262111334151786130081874972405971462972168466322891785263550418067204560531441351898508560058772475890901196825790653518404978153474717475830343232849044842574655526615395294435849865363712989044224296486388580766992201200607599937706181505634079607292843187438808974676665501849303134505458461630472422579604169924048581707174855409366303377194270140183710865132783325925884449138257983147298188639949793236413719721258966092433630859948126212337639587047675307389452018934670970187968110812115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...

4085348579... 369693080 bold digits omitted ...3322066980 digits omitted

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289

The aging of David Powers and Julia Callahan

When I initially wrote my last item about the funeral of Josephine Powers Agen, I had indicated that David Powers Sr was born ~1824, and his wife Julia Callahan, ~1826. Those dates are direct conversions from their newspaper death notices (in 1886 and 1884) of ages 62 and 58, respectively. There's always that issue of the next birthday anniversary being up-and-coming (say, in the following month) which might skew the year but the greater problem is if the reported ages actually portray any semblance of reality.

Within twelve hours of publishing that blog entry, I discovered a previously-unknown-to-me 1855 Sandy Creek NY census that in all likelihood contained our two protagonists. Julia's surname of Callahan comes from David Powers Jr's 1929 death registration wherein his parents are named. FamilySearch suggests Kallegan is the 1855 transcription of Julia's surname although, looking at the script, it might just as well be Kalligan. There's even a hint of an h in there — if that isn't an extraneous mark:

But I'm more interested in Julia's age as a means of verifying that this is our Julia Callahan. The 24 would make her birth year ~1831, five years off; not a very good fit. So why would I think that this is her? Apart from being in the right place, it's the presence in this census' family unit of an Edward Powers. Edward is noted as being in that location for only two months (compared to Julia's two years). We do not know when David Powers Sr married Julia Callahan but their first child, David Edward, was born 7 February 1857 in Sandy Creek NY, so conceived in April or May 1856. The date of the census is 8 June 1855, so Edward arrived there in March 1855. About a year to get to know and marry Julia and, nine months later, name his first-born son after himself: David Edward Powers.

If Edward is David Sr, the age of 23 would make his birth year ~1832, eight years off; an even worse fit than Julia's. Fortunately, we have both David and Julia in other censuses: 1860, 1870, 1875, and 1880.

year        David Powers Sr      Julia Callahan

1855        23   =>   ~1832      24   =>   ~1831
1860        30   =>   ~1830      30   =>   ~1830
1870        48   =>   ~1822      48   =>   ~1822
1875        50   =>   ~1825      48   =>   ~1827
1880        53   =>   ~1827      56   =>   ~1824
1884                             58   =>   ~1826
1886        62   =>   ~1824

The 1855 ages actually agree quite well with the 1860 ones. The 1870 ages appear to be ten years off suggesting a possible subtraction mistake — were it not for the continuation of the age inflation in subsequent censuses and reports at death.

As a result of contemplating all this, I edited the Josephine Agen blog entry by removing David and Julia's birth years. Much as I give credence, generally, to ages reported at death, in my database I am making their birth years 1832 and 1831, respectively, in agreement with the earlier censuses. This would make them only 54 and 53, respectively, when they died.