I did finish my computation of the 11M first items ( http://traxme.net/11m.zip first 3 are missing from the output ) which should consist of all terms under 100M. I have also looked at the problem reduced to the prime bases [2,3,5,7,11] - which has a much more pronounced holdout effect, where numbers with several smaller factors are picked up late. I have mapped the possible prime base transitions for this problem, and they form a well connected graph with lots of cycles, and crucially the graph has just a single island as opposed to the case for 4 prime bases. This indicates that all combinations of prime bases will be picked up, but with very varying probabilities. This sequence therefore shows a much wider spread: 1, 2, 6, 15, 35, 14, 12, 33, 55, 10, 18, 21, 77, 22, 20, 45, 63, 28, 40, 75, 99, 44, 50, 105, 231, 88, 80, 135, 147, 56, 100, 165, 189, 98, 110, 225, 441, 112, 160, 275, 297, 24, 70, 385, 363, 36, 140, 539, 891, 30, 175, 847, 66, 60, 245, 3773, 132, 90, 875, 5929, 176, 48, 315, 605, 242, 42, 375, 1375, 154, 54, 405, 1225, 196, 72, 495, 1715, 224, 96, 675, 1925, 308, 108, 1125, 2695, 352, 144, 525, 3025, 484, 84, 735, 6655, 704, 126, 945, 6875, 968, 162, 567, 4235, 200, 192, 693, 4375, 250, 198, 1029, 6125, 220, 216, 1323, 8575, 320, 264, 1617, 12005, 400, 288, 1089, 9317, 280, 120, 2673, 26411, 350, 150, 3267, 41503, 392, 180, 825, 65219, 448, 240, 1485, 102487, 686, 270, 1815, 184877, 784, 300, 2475, 290521, 896, 324, 1215, 9625, 616, 384, 1875, 13475, 1078, 432, 2025, 15125, 1232, 168, 3375, 33275, 1408, 252, 1575, 34375, 1936, 294, 2205, 73205, 2662, 336, 2625, 75625, 2816, 378, 2835, 166375, 3872, 486, 1701, 18865, 440, 576, 3087, 21175, 500, 648, 2079, 21875, 640, 396, 2541, 30625, 800, 528, 3969, 42875, 550, 594, 5103, 60025, 880, 726, 7203, 84035, 1000, 768, 3993, 29645, 490, 864, 8019, 46585, 560, 972, 9801, 48125, 700, 1152, 11979, 67375, 980, 1296, 24057, 94325, 1120, 1458, 29403, 105875, 1250, 1536, 4851, 109375, 1280, 792, 6237, 153125, 1600, 1056, 7623, 214375, 2000, 1188, 9261, 300125, 1100, 1452, 11907, 420175, 1210, 1584, 15309, 546875, 1760, 1728, 21609, 132055, 2200, 1944, 27783, 148225, 2420, 2304, 35721, 207515, 2500, 2592, 11319, 171875, 2560, 504, 14553, 366025, 3200, 588, 17787, 378125, 4000, 672, 18711, 805255, 5000, 756, 22869, 831875, 5120, 882, 27951, 859375, 6250, 1008, 33957, 1830125, 6400, 1134, 35937, 1890625, 1400, 1176, 43923, 4026275, 1750, 1344, 72171, 4159375, 1960, 1512, 88209, 4296875, 2240, 1764, 107811, 8857805, 2450, 2016, 131769, 9150625, 2800, 2058, 216513, 9453125, 3430, 2268, 264627, 20131375, 3500, 2352, 323433, 20796875, 3920, 2646, 395307, 21484375, 4480, 2688, 483153, 44289025, 4900, 2916, 649539, 232925, 1372, 3072, 3645, 240625, 1568, 3456, 4125, 326095, 1792, 3888, 4455, 336875, 2744, 4374, 5445, 456533, 3136, 360, 7425, 717409, 3584, 450, 9075, 1127357, 4802, 480, 12375, 1294139, 5488, 540, 13365, 2033647, 6272, 600, 16335, 3195731, 7168, 720, 19965, 5021863, 9604, 750, 20625, 7891499, 10976, 810, 22275, 9058973, 12544, 900, 27225, 12400927, 14336, 960, 37125, 14235529, 19208, 1080, 40095, 22370117, 21952, 1200, 45375, 35153041, 25088, 1350, 49005, 55240493, 28672, 1440, 59895, 63412811, 33614, 1500, 61875, 86806489, 38416, 1620, 66825, 99648703, 43904, 1800, 81675, 136410197, 50176, 1920, 99825, 156590819, 57344, 2160, 103125, 246071287, 67228, 2250, 111375, 386683451, 76832, 2400, 120285, 443889677, 87808, 2430, 136125, 607645423, 100352, 2700, 147015, 697540921, 114688, 2880, 179685, 954871379, 134456, 3000, 185625, 1096135733, 153664, 3240, 200475, 1500512167, 175616, 3600, 219615, 1722499009, 200704, 3750, 226875, 2706784157, 229376, 3840, 245025, 3107227739, 235298, 4050, 299475, 4253517961, 268912, 4320, 309375, 4882786447, 307328, 4500, 334125, 6684099653, 351232, 4608, 5625, 471625, 1694, 5184, 6075, 512435, 2156, 5832, 9375, 529375, 2464, 6144, 10125, 588245, 3388, 1782, 10935, 765625, 4312, 2112, 16875, 1071875, 4928, 2178, 18225, 1500625, 6776, 2376, 28125, 2100875, 7546, 2904, 30375, 2734375, 8624, 3168, 32805, 2941225, 9856, 3564, 46875, 3828125, 11858, 4224, 50625, 4117715, 13552, 4356, 54675, 5359375, 15092, 4752, 84375, 7503125, 17248, 5346, 91125, 10504375, 18634, 5808, 98415, 13671875, 19712, 6336, 140625, 14706125, 23716, 6534, 151875, 19140625, 27104, 6912, 164025, 660275, 5324, 7776, 3675, 741125, 5632, 8748, 4725, 924385, 7744, 9216, 5145, 1037575, 10648, 10368, 6615, 1164625, 11264, 11664, 7875, 1203125, 15488, 12288, 8505, 1452605, 21296, 13122, 11025, 1630475, 22528, 13824, 13125, 1684375, 29282, 15552, 14175, 2282665, 30976, 17496, 15435, 2358125, 42592, 18432, 18375, 2562175, 45056, 20736, 19845, 2646875, 58564, 23328, 23625, 3301375, 61952, 24576, 25515, 3587045, 85184, 26244, 25725, 3705625, 90112, 27648, 33075, 4621925, 117128, 31104, 36015, 5187875, 123904, 34992, 39375, 5636785, 170368, 36864, 42525, 5823125, 180224, 39366, 45927, 6015625, 2750, 41472, 50421, 6470695, 3520, 46656, 64827, 7263025, 4400, 49152, 83349, 8152375, 4840, 52488, 107163, 8421875, 5500, 55296, 137781, 10168235, 6050, 62208, 151263, 11413325, 7040, 69984, 194481, 11790625, 8000, 73728, 43659, 12810875, 10000, 78732, 53361, 13234375, 10240, 82944, 56133, 15978655, 12500, 93312, 68607, 16506875, 12800, 98304, 79233, 17935225, 16000, 104976, 83853, 18528125, 20000, 110592, 101871, 20588575, 20480, 7128, 124509, 26796875, 25000, 7986, 130977, 28824005, 25600, 8448, 160083, 37515625, 31250, 8712, 168399, 52521875, 32000, 9504, 195657, 68359375, 40000, 10692, 205821, 73530625, 40960, 11616, 237699, 95703125, 50000, 12672, 250047, 102942875, 8800, 13068, 321489, 133984375, 9680, 14256, 352947, 144120025, 11000, 15972, 413343, 187578125, 12100, 16038, 453789, 201768035, 13310, 16896, 583443, 262609375, 13750, 17424, 750141, 341796875, 14080, 19008, 964467, 367653125, 17600, 19602, 1058841, 478515625, 19360, 21384, 1240029, 514714375, 22000, 23232, 1361367, 669921875, 24200, 23958, 1750329, 720600125, 26620, 25344, 2250423, 937890625, 27500, 26136, 2470629, 1008840175, 28160, 28512, 2893401, 1313046875, 30250, 31944, 3176523, 1412376245, 35200, 32076, 3720087, 1708984375, 38720, 33792, 4084101, 1838265625, 44000, 34848, 5250987, 2392578125, 48400, 38016, 6751269, 2573571875, 51200, 39204, 251559, 3349609375, 62500, 42768, 305613, 3603000625, 64000, 46464, 307461, 4689453125, 80000, 47916, 373527, 5044200875, 81920, 48114, 392931, 6565234375, 100000, 50688, 480249, 7061881225, 102400, 52272, 505197, 8544921875, 125000, 57024, 554631, 9191328125, 128000, 58806, 586971, 9886633715, 156250, 63888, 617463, ... Up and to this point 3024, 3402, 3528, 4032, 4116, 5636, ... are the smallest candidates that have not yet been included. But I see very little reason to believe that they will be left out, based on how the process appears to have good quality randomness. And A336957 could also be expected to have the same property of including all possible composite numbers, as the number of bases is infinitely larger. Cheers, Frank On Sat, Aug 22, 2020 at 11:38 PM christopher landauer <topcycal@gmail.com> wrote:
hihi, all -
one of the things i noticed is that a very large fraction (i know better than to say almost all in this group 8-)) of the values a[n] are ``late'', where late means a[n] < n
the opposite ones are early, like 6 and 15
but most of the late ones are not very late, usually within a few percent of n (at least up to n=10^6, and the percentage is also decreasing, irregularly as seems to be the case with all properties related to primes) - i wonder of some kind of density argument might get somewhere
it is also interesting to look at the sequence gcd(a[n],a[n-1]) for n>2, which has a subsequence that seems to be all the primes, all in order (but not consecutive; there are lotsa pesky 2's) with a density that appears to be approaching 0.5; that might be a useful step, to show that every prime p occurs in some a[n] with n < 2*p or so
more soon,
chris
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