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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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.

The funeral of Josephine Powers Agen

The chart shows what I currently know about the shoots of my wife's Powers ancestry. At the top are David Edward Powers (1857-1929) and Agnes Sullivan (1857-1936), her great-grandparents. David Edward's parents, David Powers Sr (died 1886) and Julia Callahan (died 1884) were alone at the root of the tree until March 2011, when my research associates and I discovered a sibling for David: Michael Powers (1845-1918) with wife Mary Brady (1847-1886).

Michael's eldest child, Josephine, died 22 Dec 1920 and the funeral was held two days later. A Christmas eve Utica Herald-Dispatch report named the pallbearers: John Daunt of Lacona, J.S. Powers of Watertown, James B. McIntosh of Albany, James and John Powers of Utica, Edward F. Dunn of Whitesboro. Five of the six names are in the chart and provide a wonderful means to tie the family units together. But who was James Powers of Utica?

Subsequent searches failed to identify such a family member, much as we might have liked to discover yet another branch in the tree. My best guess is that the newspaper put James and John together because they were both from Utica and then (for an unknown reason) omitted James' surname. So, probably, James Powers was in fact James E. Costello, whose family is the most likely of the three that are without a pallbearer representative to have contributed one.

Guerilla street-spam fighter

On 17 April 2013, I requested from my local municipal councillor, Frances Nunziata, that some street-spam be removed from my residential neighbourhood. The councillor's administrative assistant, Ji Na Park, said they would refer my complaint to the Municipal Licensing and Standards department. "The signs are removed from the posts and the companies are contacted to be advised that this should not continue."

On 2 May 2013, I asked: "What I would like to know now is what sort of time-frame I should expect for your implied action to take place (if, that is, it will ever take place), since two weeks has proven insufficient for it. Perhaps you were mistaken in thinking that the Municipal Licensing and Standards department actually does this." Ji Na replied: "We apologize for the delay. As the signs are considered non-urgent, they are classified as non-priority and may take a number of weeks."

Two days later I removed 30 of the signs myself. I informed the councillor's office but added: "There were three signs that I have been unable to remove because they were too high up, and seriously bolted into the telephone poles (not just nailed)... May I have the city's co-operation in having these three signs removed?" Ji Na's 7 May 2013 reply: "We have reported in the 3 signs to Municipal Licensing and Standards for removal. Please let us know when you see improvements. As it is peak season, there will likely be more signs going up in the future. We will try and get them removed as soon as possible."

Today, after 55 weeks of waiting, I asked Catherine's brother (here for a few hours en route to England, leaving his truck in our yard) to help transport a ladder to the crime scenes. In the photo you see me, utility knife in hand, evaluating the job to be done:

Alternating sign

Simple continued-fraction representations of constants are a source of much mathematical amusement. The picture is a scan of the cover of Carl Douglas Olds' 1963 classic Continued Fractions which served as my introduction to the topic around 1967.

The sequence [3; 7, 15, 1, 292, 1, 1, ...] is of course the start of the continued-fraction expansion for π. I wish C.D. Olds had used (and insisted on) that semicolon (instead of a comma) after the initial 3. The book shows us how to calculate from these numbers convergents to π, numerators and denominators of the convergents being derived separately. For numerators we pretend things start with [0, 1, ...] and for denominators, [1, 0, ...]. The numerator- and denominator-sequences are then calculated by taking an entry from the continued-fraction sequence (starting with 3), multiplying it by the previous number, and adding the previous-to-previous number.

So, for numerators: [0, 1, 3*1+0 = 3; 7*3+1 = 22, 15*22+3 = 333, ...]
For denominators: [1, 0, 3*0+1 = 1; 7*1+0 = 7, 15*7+1 = 106, ...]

Dropping our pretend prepends and dividing: [3/1; 22/7, 333/106, 355/113, ...]. These are the well-known convergents to π. Note that the convergents alternate in their relation to π: [<π; >π, <π, >π, ...]. Subtracting π: [<0; >0, <0, >0, ...]. Or, if you will, their sign: [-1; 1, -1, 1, ...].

One can do the same thing for τ (pronounced tau), π's less well-known parent (more commonly referred to as 2π). The continued-fraction expansion for τ is [6; 3, 1, 1, 7, 2, 146, ...]. The convergents are: [6; 19/3, 25/4, 44/7, ...]. Like π, these convergents alternate in their relation to τ: [<τ; >τ, <τ, >τ, ...], their sign likewise.

What happens when one divides the convergents to τ by the convergents to π? The resulting sequence is [2; 133/66, 1325/666, 4972/2485, ...], numbers very close or equal to 2. To generate their sign, subtract 2: [0; 1/66, -7/666, 2/2485, ...], hence [0; 1, -1, 1, ...]. The twist is that the sign sequence is now littered with zeros (in seemingly random — but decidedly clustered — locations). How do those zeros affect the sign alternation? The surprising (to me) answer is: not at all. Remove all zeros from Convergents[τ]/Convergents[π]-2 and what remains will alternate sign, regardless of the number or placement of the previously intervening, single-or-consecutive zeros.

Zipp

A photo of my new Zipp speaker, unzipped.

A long time ago I had set up speaker wires to two ceiling corners of the kitchen to which were attached small, inexpensive (but effective) Radio Shack speakers. An alternate pair of wires had gone (via a different route) outside the house and ended up on the eave soffit by the back deck. I could detach the speakers from their brackets in the kitchen and attach them to the brackets outside.

The outside wires were removed I-can't-remember-when and the inside speakers came down last year when the kitchen was "refreshed". To listen to music on the back deck now meant turning up the volume in my adjacent computer-room and leaving the window wide open.

But that volume was now coming from my iMac speakers, because when it replaced my old under-the-desk Power Mac G5, I lost the optical-out connection to my five-speaker Logitech setup. Yea, I know: I needed a minijack-to-optical inbetweener, but I don't get out much. :P

Anyways, I finally ordered a Belkin Toslink cable that came with that inbetweener and ordered the Zipp speaker at the same time. So I can listen to music again on my Logitech speakers in my room, and on the Zipp outside — or anywhere else within my local-network WiFi reach.

The French connection

Catherine's father appears not to have been particularly interested in his parents' family connections, a point that was driven home to me last year when I discovered that his mother had (until 1940) a sister living in Winnipeg, of whom he himself (when I asked him about her) seemed unaware. I have since come into possession of a good number of handwritten letters from that sister to his mother, as well correspondence from France and newspaper articles that have (collectively) allowed me to flesh out some of the history.

Jean Baptiste Homard (born 1852), a "well-known contractor in northern France", married Marie Claire Vassel (born 1857) and they had (in 1879) a daughter, Marie Isabel [Jeanne] (anglicized to Mary Elizabeth [Jane], that unknown sister). Jean Baptiste and Marie Claire were on the steamer Labrador, arriving in New York (from Havre) on 28 Jul 1886. A daughter, Emélie Matilde (anglicized to Emily May, Catherine's grandmother), was born 1 Apr 1887 in Fort William, Ontario. Another daughter, Gabrielle, was born in 1889 — supposedly back in France. Marie Claire's mother, Catherine Bouquet Vassel (born in 1826) ended up back in Canada with them in 1891 (according to the 1911 census). Gabrielle died of scarlet fever (in 1903) in St. Norbert, Manitoba.

Jane had married Albert George Cowley in 1898 in Winnipeg and they ended up with three children: two boys with mostly-known, living descendants and a girl without offspring. Here they are (sans daughter, plus pet dog) circa 1910:

By the time Emily married Frank Powers (3 Jul 1912) in Calgary, Alberta, Jean Baptiste and Marie Claire Homard, and Catherine Vassel, had returned to France — to Thiaucourt. At the start of World War I, they became prisoners-of-war, ending up in Frankfurt-am-Main, Germany. Catherine died a year or so into her incarceration. Marie Claire and Jean Baptiste survived "well, but hungry" and were released at the war's end. Marie Claire died 12 Dec 1923 and Jean Baptiste, 5 Apr 1924. An 8 Apr 1924 letter to Emily from a cousin, Jeanne Vautrin in Thiaucourt, gives details of Jeanne's family, including a photograph circa 1920:

The Vautrin children are Jean (born ~1906), Michel (born ~1913), and Marie Blanche (born ~1915). The masculine-faced, dark complexioned adult in the photo must be Jeanne because her husband, Alfred François Vautrin (1 May 1880, Beney-en-Woëvre - 20 Aug 1914, Bréhain), would have been dead. Jeanne also mentioned a sister, Marie. Chances are good that there are living descendants in that part of France.

Canoeing in Slocan Lake

In July 2004 we ended up in British Columbia for a week. The photo shows Catherine just after noon on the 21st as we leave New Denver to paddle across Slocan Lake. Visible on the other side are some of the peaks of northern Valhalla Provincial Park, including the small glacier of Mount Denver. We pitched a tent at Wee Sandy Creek beach, spent the night, and on the following day paddled down to Nemo Creek beach, where we set up and spent another two nights before heading back to New Denver. A particular highlight was this waterfall on Nemo Creek:

I mention this, now almost ten years later, because of a just-happened, tragic New Denver canoeing mishap.