## Tuesday, February 28, 2023

### A million-digit Leyland prime (start of a new run)

Five days ago I started a new run of testing candidates for the property of being a million-digit Leyland prime. The million-digit part is relatively easy; the primality testing, not so much. My last run required nine months, not counting the month it took to sieve. I'm not pushing it for now. Of the 59364 candidates in this run, I'm only doing 9000 on my three iMacs. This should be done in July. My Mac minis are testing much smaller Leyland numbers (currently ~386750 digits) for primality and I'll keep that going until I exhaust my current crop of sieved numbers in that range. Only then will I divert them to help in the million-digit hunt.

## Monday, February 13, 2023

### Small-string final non-appearance coincidences in base-ten powers of two

If we look at OEIS A094776, one sees the beginnings of sequences that apply the inherent concept to strings of more than one digit. Keith Lynch suggested the idea (tongue-in-cheek, I thought) on MathFun a few days ago and Maximilian Hasler actually worked out the numbers for strings 10 to 18. I decided to chart a more comprehensive listing...

My results: sorted by strings and sorted by powers of two. There are "coincidences" where two or more strings share the same power of two as their final non-appearance exponent (for example, the two 71s in A094776 for digits 5 and 7). I'll list those here after brief summaries of each n-digit result:

1-digit strings

{71,5}
{71,7}
...
range: 119.5 ± 48.5
average: 1026/10
...
{153,3}
{168,2}

string coincidence

71: {5,7}

2-digit strings

{1300,91}
{1416,07}
...
range: 2399.5 ± 1099.5
average: 215386/100
...
{3493,28}
{3499,95}

string coincidence

2146: {33,48}

3-digit strings

{20589,141}
{20729,713}
...
range: 37290.5 ± 16701.5
average: 28860154/1000
...
{51375,552}
{53992,661}

string coincidences

22044: {024,275}
24486: {404,675}
25305: {410,947}
25440: {317,604}
25668: {442,815}
25704: {123,766}
25980: {096,868}
26046: {378,588}
26136: {422,677}
26316: {227,929}
26477: {152,690}
26695: {085,256}
26792: {048,732}
27003: {737,974}
27121: {545,932}
27479: {183,687}
28196: {300,554}
28252: {116,641}
28270: {099,575}
28317: {578,656}
28425: {287,392}
28532: {171,910}
28609: {017,919}
28784: {033,719}
28850: {164,647}
28891: {346,505}
29173: {648,787}
29705: {668,997}
29711: {335,799}
29976: {665,995}
30977: {131,395}
32637: {076,426}
33550: {555,796}
33607: {582,598}
33631: {117,735}
39571: {021,622}

4-digit strings

{266774,6355}
{273317,4949}
...
range: 459917 ± 193143
average: 3642758125/10000
...
{625874,6636}
{653060,6164}

string coincidences

296341: {0317,9730}
299582: {2041,2103}
302068: {8362,9844}
303160: {1883,2335}
304631: {5262,7194}
304639: {2896,5816}
305663: {5142,5397}
307552: {0904,8968}
307739: {9064,9656}
308220: {2396,6102}
308408: {5509,8025}
308917: {0658,5181}
309422: {2879,8852}
310452: {2800,6576}
310789: {0093,5900}
311943: {4436,7337}
312480: {5954,9044}
312956: {1741,1944}
313628: {7855,9582}
314127: {4003,6689}
315423: {6059,6252}
315844: {3119,7155}
316864: {2825,4655}
317042: {0507,0655}
318106: {7886,9390}
318428: {1872,4548}
318594: {5091,5501}
318703: {0505,6814}
319722: {7009,7541}
320081: {0089,1201}
320549: {4780,9688}
321025: {0020,8594}
321026: {1045,1488}
321108: {2364,3963}
322286: {5390,9872}
322907: {5976,9894}
323025: {3082,5477}
323178: {4438,7411,9268}
323872: {5264,8601}
323973: {7651,9051}
324814: {4462,4521}
324885: {4116,8104}
326420: {5641,8345}
326585: {0703,3413}
326793: {7640,9369}
326886: {3632,7622}
327005: {7087,8412}
327241: {0013,4113}
327387: {2757,6279}
327406: {8469,9238}
327692: {0328,1158}
327856: {6037,7447}
327899: {2569,8404}
328022: {2068,8665}
328433: {3715,9685}
328441: {4541,5491}
328592: {1770,7600}
329117: {2725,3266,7959}
329142: {1846,7235}
329285: {0324,7893}
329773: {1510,6807}
329834: {3380,5873}
330434: {5197,8702}
330442: {0232,8274}
330690: {4111,6761}
331061: {3225,5929}
331358: {2540,7253}
331463: {1493,9185}
331571: {1174,7276}
331610: {8690,9273}
332045: {1192,9701}
332136: {2114,8049}
332481: {1728,5967}
332507: {4609,5093}
332627: {0126,2700}
333146: {2636,5860}
333450: {8609,9537}
333462: {5600,8478}
333663: {1050,7693}
333682: {8270,9353}
333998: {4034,7687}
334303: {3254,4714}
334617: {6586,7828}
334837: {2030,8091}
334914: {9034,9881}
335184: {3039,6080}
335216: {5066,9474}
335580: {4004,4864}
335618: {5891,9991}
335647: {4715,9359}
335648: {1060,3150}
335833: {0604,9269}
335867: {5145,8779}
336146: {7404,8354}
336846: {2783,4059}
337394: {0750,7595}
337524: {5159,9199}
338108: {4509,6866}
338157: {2741,4933}
338377: {1494,2094}
338420: {1515,3966}
338794: {9594,9845}
339154: {0743,9414}
339356: {1476,5817}
340009: {2933,8133}
340042: {2953,9957}
340097: {0172,1204}
340281: {5404,8419}
340472: {0060,3472}
340926: {0872,6788}
341131: {0237,6434,8154}
341157: {1810,9578}
341370: {5676,5786}
341506: {1432,6014}
341528: {3645,7689}
341902: {2203,6407}
341914: {5194,8521}
341983: {6118,7811}
342019: {2264,5218}
342233: {3949,5461}
342260: {0897,8605}
342394: {0260,8494}
342564: {4002,9173}
342630: {5527,8773}
342757: {3569,9032}
342821: {4799,5993}
343798: {7343,8491}
344003: {5961,7381}
344041: {2518,3260}
344251: {0102,3745}
344396: {1523,4810}
344780: {9627,9882}
345245: {1558,7972}
345278: {8163,9325}
345602: {4701,5555}
345933: {4788,9070}
345936: {1277,4166}
346134: {2839,8391}
346151: {0856,3654}
346219: {5495,5683}
346364: {4192,7639}
346769: {0587,1635,2374}
347036: {0559,8406}
347161: {8907,9741}
347522: {3605,3996}
347620: {2511,6244}
347740: {0579,8606}
347746: {4358,8878}
347775: {8741,9906}
347907: {2204,9640}
347925: {5423,6137}
348332: {6774,8667}
348762: {3488,6962}
348855: {0956,2340}
348911: {3646,9581}
349048: {3546,3744}
349073: {4179,8338}
349254: {3805,6726}
349471: {1047,4986}
349829: {6157,7860}
350183: {2142,6961,9708}
350308: {4130,6670}
350516: {4296,8527}
350569: {4060,9635}
350635: {1378,2927}
350684: {1261,7773}
351243: {1769,3536}
351326: {3121,7915}
351529: {2380,6571}
351561: {3911,5291}
351713: {2415,8585}
352134: {0692,1374}
352235: {5647,6629}
352294: {7046,8069}
352417: {0619,5419}
352438: {2439,8199}
352539: {6748,8705}
352694: {1180,7163}
353011: {2131,6621}
353082: {0174,4600}
353168: {1889,5682}
353239: {4050,9956}
353464: {2294,4966}
353828: {4931,7372}
353840: {0217,0907}
354455: {3776,6185}
355087: {4417,6577}
355188: {3510,5193}
355324: {0017,2318}
355402: {6057,7503}
355434: {3307,6790}
355805: {2138,3581}
355947: {1746,6743}
356094: {2159,6092}
356103: {2035,2615}
356255: {7001,8420}
356497: {1586,2339}
356516: {0066,0762}
356620: {3657,4174}
356900: {2055,5223}
356973: {8776,9103}
357223: {1687,7664}
358009: {7382,7497}
358487: {1366,4133}
359062: {1146,8147}
359220: {3660,8822}
359230: {6165,8191}
359616: {0633,6176}
360109: {2307,7362}
360826: {3455,7078}
361585: {4828,9217}
361678: {3837,7685}
362060: {0732,5850}
362094: {4434,8748}
362106: {2659,8340}
362129: {1302,2693,3550,9945}
362530: {2483,3015}
362580: {1281,5708}
362896: {4255,7722}
362924: {4744,9644}
363104: {0674,3672}
363222: {3484,7142}
363547: {2239,5260}
363885: {6291,7596}
363897: {2848,4773}
364027: {1035,8633}
364036: {7200,8926}
364503: {0589,5054}
364986: {0798,8765}
365010: {0005,8973}
365249: {5474,7459}
365443: {0073,7884}
365632: {1824,7237}
365776: {0071,8921}
365966: {0244,7492}
365968: {4984,5128}
365996: {7230,9404}
366225: {1300,3345}
366355: {9461,9915}
367072: {1481,5969}
367111: {2634,6306}
367217: {2393,2429}
367898: {4498,8180}
368443: {5085,7256}
368694: {5507,8283,9984}
368976: {6996,7062,7667}
369313: {2716,8640}
369495: {4545,7392}
369970: {2764,5011}
370814: {4601,6747}
370974: {4575,5613}
371553: {2804,6497}
372064: {6702,8361}
372108: {2949,7989}
372252: {1248,7907}
372311: {0050,5804}
372356: {4343,8399}
372357: {1716,7165}
372814: {6287,7363}
373523: {4530,8105}
373861: {3379,4697}
374797: {5925,8296}
374810: {1433,9562}
375824: {2284,6196}
376003: {1052,8083}
376138: {4294,7424}
376739: {9078,9973}
376760: {2051,9750}
376785: {0438,7605}
377455: {4277,5315}
378037: {2196,7421}
378355: {2938,9245}
378973: {2147,9852}
378984: {0276,6941}
379462: {0014,8562}
379612: {0680,5393}
379782: {1791,2266}
380544: {1328,8863}
381077: {7870,9296}
381177: {2894,7066}
381669: {0006,5820}
381849: {0597,5625}
382182: {3690,4675}
382940: {3346,8450}
383096: {4242,6254}
383242: {2584,7346}
383783: {3804,9176}
383870: {3557,9628}
384280: {1068,7269}
384385: {2436,4817}
384511: {3194,6645}
384649: {0881,6310}
385168: {4726,5116}
385543: {2545,8762}
386204: {9665,9773}
386294: {0727,1169}
387984: {5505,9448}
388202: {5985,8507}
389330: {1857,9447}
389349: {2841,5269}
389481: {2729,3998}
389795: {2367,4654}
389853: {1809,9849}
390037: {5162,5844}
390708: {0628,9706}
391278: {5899,7553}
391837: {2013,8526}
391929: {1713,3793}
392002: {2034,2872}
392518: {7118,9658}
393878: {5496,9517}
396142: {6936,8249}
396567: {7081,8211}
396886: {3938,8909}
397399: {0352,2680}
397503: {5059,9082}
397631: {1691,7111}
398347: {1367,5590}
398451: {2077,3116}
398583: {0687,9055}
398635: {4387,6361}
401346: {0266,4432}
402139: {5528,6893}
402315: {0936,7464}
403677: {3809,8263}
405493: {6027,6698}
407926: {0891,6640}
412908: {0278,3055}
413260: {0749,9602}
413732: {6654,7788}
414179: {8067,8580}
415732: {0870,6078}
416017: {0320,5720}
417139: {1126,9616}
417664: {4785,6195}
419623: {3402,3883}
420795: {1676,6722}
421861: {0786,1291}
423699: {3309,8888}
426750: {7638,9766}
433287: {1151,7375}
441637: {2651,2822}
452674: {3564,8663}
504659: {3418,3774}
514562: {0052,1787}

The red numbers in the summaries for each n-digit result has become a sequence: A360623.

## Tuesday, February 07, 2023

### Fenestron

The loud sound of a helicopter just before 10:30 p.m. last night got me to look out the window and, because of the unusual appearance of the tail, run to get my camera and take a photo from the front porch. The aircraft disappeared behind the apartment buildings on Weston Rd. — seemingly landing (in the vicinity of Weston's UP Express train station; I'm going to guess at the Toronto Paramedic Services lot). Coincidentally, a UP Express service alert appeared one-and-a-half hours later.

## Friday, February 03, 2023

### Lake effect

 click to enlarge
It is sunny against a mostly blue sky backdrop outside but, as predicted, very cold. I did not even take Bodie for his walk this morning! My GOES 16 screen grab (above) shows snow streamers coming off the Great Lakes (left) with the prevailing northwesterly winds (below). The situation is even more pronounced off the coast of Maine (far right) where the lake effect is an ocean effect.
 click to enlarge