## 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.