Tuesday, July 22
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.
Tuesday, July 15
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.
Wednesday, July 2
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.)
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.)
Friday, June 27
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, 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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.
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, 644276, 644279, 644423, 644426, 644429, 644453, 644456, 644459, 644483, 644486, 644489, 644513, 644516, 644519, 644573, 644576, 644579, 644723, 644726, 644729, 644753, 644756, 644759, 644813, 644816, 644819, 644843, 644846, 644849, 644873, 644876, 644879, 645113, 645116, 645119, 645173, 645176, 645179, 645713, 645716, 645719, 645773, 645776, 645779, 647123, 647126, 647129, 647153, 647156, 647159, 647213, 647216, 647219, 647243, 647246, 647249, 647513, 647516, 647519, 647723, 647726, 647729, 647753, 647756, 647759, 648113, 648116, 648119, 648173, 648176, 648179, 648413, 648416, 648419, 648443, 648446, 648449, 648473, 648476, 648479, 648713, 648716, 648719, 648773, 648776, 648779, 651113, 651116, 651119, 651173, 651176, 651179, 651713, 651716, 651719, 651773, 651776, 651779, 652223, 652226, 652229, 652253, 652256, 652259, 652523, 652526, 652529, 652553, 652556, 652559, 652823, 652826, 652829, 652853, 652856, 652859, 652883, 652886, 652889, 655223, 655226, 655229, 655253, 655256, 655259, 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, 715157, 715167, 715217, 715227, 715237, 715247, 715287, 715297, 715317, 715357, 715367, 715517, 715527, 715557, 715567, 715597, 715627, 715637, 715647, 715657, 715667, 715697, 715917, 715927, 715937, 715997, 716117, 716127, 716157, 716217, 716227, 716257, 716327, 716337, 716357, 716367, 716397, 716417, 716427, 716487, 716557, 716617, 716627, 716637, 716647, 716697, 716817, 716827, 716847, 716857, 716887, 716917, 716927, 716957, 716967, 716997, 719127, 719157, 719217, 719227, 719287, 719337, 719357, 719367, 719517, 719557, 719617, 719627, 719637, 719647, 719657, 719687, 719917, 719927, 719937, 719967, 719997, 721117, 721127, 721157, 721167, 721197, 721227, 721237, 721247, 721257, 721267, 721297, 721317, 721327, 721337, 721367, 721397, 721517, 721527, 721537, 721597, 721617, 721647, 721657, 721667, 721687, 721927, 721937, 721957, 721967, 721997, 722137, 722157, 722167, 722217, 722227, 722237, 722247, 722297, 722317, 722357, 722367, 722417, 722427, 722437, 722447, 722457, 722487, 722497, 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.
Thursday, June 26
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.
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.
Saturday, June 14
9^9^9^9
Evaluated of course from the top down: 9^(9^(9^9)) =
21419832947968056113333643734424808301472270728451284887065161959828087496567048470361184472499173685348825764518319411249675059163057939450132383137857257387303899906076223751642158506081537516902249534368182485563436460743204914221083241359509171979501742403737274498190992986234420741963952331470128519454586086183389560459447508519174567154661923513414667736134400991225435672831447817195155379261398749901736062235194235305257317470352535034598833851235160105776671276946104531563857467658976215624043993795779561336003808688926266884283761352441497964782806972761629249342998478005772448909628121104203754931573509482876386281192028953230881806350058010597823777723094132626264578746316256673928426734253433557310495252276792426000946342114034655162413993889346040025558488501379489211584073640000670401588755083736879801550128723434452732373594188602708684863434667860439349440029670400440211337834144962661525484940521999586681805342308223638014652970224685931565568787952219583051583760692665335728780491120909774512599634140270525211434950910562004862991500563860362784147113854804111721510669363426192871540771532588537785733596195270893543172952866225695049165504170200387017043648598016350921587164435363590773303193710778281500541706099414499374130783289007571097319991330040793257406144794022866312593734363512505028018181757699431623730362825141400084201377732055413184733884246186781690131481032352870071901478070878002728132575826640389832532452337525474162592160525288746795727173901735308910110418559632856038429026600648046284160478291787441379326887894347380169206804072022656881349795957890411151700619897854232776881602660007591495541686357630643413099875580703711848095192915749513710023681426562083696413249244304846933560112450385621972240709351464870723879171250706461876757276382612247791670102965366491521477461192657286286686997435699646809578084008568030860483298051587062407328939846546044851729698471329507733283005454052568763236711889000213264716126852718152634154278324290850376993674713864120455120213670722146476721225325078575154901670322007726387082586092730520192259534163815879741736191752594107802548859899652249831169519173588631223437682950230138533178252678097248936287545550539531790153295476611233774538079213115131372514203838588023224526073699784338233113989108244605285009206161082435622947620678974760588393116114349053742518934087148032516574282470515947852012334409693853256811465531586246665952617532044489026105586454007354500639698864881481870834540293047438520744225330668259771869163603116541965700588490445673431619614762936509103240863885315919317042736999706475202464717014802207877539565433704962457775870709201947013460276420454955168669179942404186367520759098483957187057667687422498148163938227735579656358915661264479854579436881770645936131226513142519724463042874418807189979874674266750389040842107384189613419938737245821675358400872657469614579704360701127366825258542791295924213240392280570173797074000109966557177865975907103462288638372720801043868399220499501845883108175247867245503293313884441066362667722424940398617732320900366045529821408386312046037549299194758531800396171364286076210238412059277853224491402153094099140931845421823330466494048710696221318746867639564959692431682505296039969354814380499445300800917574017198570878762399458795171318508568332984880615336673462004387365505518127371012997914806708131522105101294255628149198107105629222041915747683741984328947471716000663006172812933414770723169962371393963681187382384701871358232965345968827185739888932658209796139587407752076843272017801027363325505620993374150100380345978437169022059348662355392750543962418176135982953929193245881871992577351288227199525525672162777037369779993842507258983012507005230908194449478278767115369152101406777750616299533255032976571201848676526916238407266722539339703860141178186954328277654012563050254381444509878281404729668264995808316604156288715005059915700303502946028741087979953976431522765778877235981384740818778166658825968527102178265345054485558256232242739367400360374491226506588072144365872765202068905635109399653350047803247595332564428122409532102816104552855202153189018603947244297559027570339268573460527499371751164149140297165476333046781060914784845729334021263001078166991739266659151865514344968369470546121206451208260149094271086957771896787390624887863768384968986158459649965518375616802530458614364453185879824811106315439747116815133445128326680457044152404890438589029224177709276262111334151786130081874972405971462972168466322891785263550418067204560531441351898508560058772475890901196825790653518404978153474717475830343232849044842574655526615395294435849865363712989044224296486388580766992201200607599937706181505634079607292843187438808974676665501849303134505458461630472422579604169924048581707174855409366303377194270140183710865132783325925884449138257983147298188639949793236413719721258966092433630859948126212337639587047675307389452018934670970187968110812115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Friday, June 13
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.
Tuesday, June 10
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.
Tuesday, May 27
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:
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
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.
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