View Single Post
Old 2021-04-07, 10:20   #1074
garambois
 
garambois's Avatar
 
"Garambois Jean-Luc"
Oct 2011
France

15518 Posts
Default

One new conjecture


This new conjecture completes and replaces conjectures (134) and (135).
k is an integer.


Conjecture (140) :
If a base b = p# is primorial (p prime > 41), then s(b^(2 + 6 * k)) is abundant.


Let me present you the data that allowed me to state this conjecture, then give you some explanations and finally ask you a curious question following a curious observation, in bold at the bottom of the post (paragraph "For b = 3# to b = 5#").


Data (Tables not complete presented here, because too long) :

Code:
b = 3# = 6 = 2 * 3
[720, 1080, 1260, 1440, 1680, 1800, 1980, 2160, 2340, 2520, 2640, 2700, 2772, 2880, 3120, 3240, 3360, 3600, 3780, 3960, 4032, 4200, 4320, 4620, 4680, 4752, 5040, 5280, 5400, 5460, 5760, 5880, 5940, 6120, 6240, 6300, 6480, 6552, 6720, 6840, 7020, 7200, 7392, 7560, 7920, 8100, 8280, 8400, 8640, 8820, 9000, 9072, 9240, 9360, 9720, 9900, 10080, 10440, 10560, 10800, 10920, 11340, 11520, 11760, 11880, 12240, 12480, 12600, 12960, 13104, 13440, 13680, 13860, 14040, 14400, 15120, 15840, 16200, 16380, 16560, 16800, 17280, 17640, 17820, 18000, 18360, 18480, 18720, 18900, 19440, 19800, 20160, 20520, 20880, 21060, 21120, 21600, 21840, 22680, 23760, 24300, 24840, 25200, 25920, 26460, 27000, 27720, 28080, 29160, 29700, 30240, 31320, 31680, 32400, 32760]

b = 5# = 30 = 2 * 3 * 5
[55440, 65520, 75600, 80640, 90720, 100800, 110880, 120960, 131040, 138600, 151200, 166320, 171360, 181440, 196560, 201600, 221760, 226800, 241920, 262080, 272160, 277200, 302400, 332640, 342720, 362880, 393120, 403200, 415800, 453600, 498960, 514080, 544320, 589680, 604800]

b = 7# = 210 = 2 * 3 * 5 * 7
[80, 440, 1520, 1700, 1808, 2240, 2960, 3248, 3680, 3968, 4400, 6560, 6848, 7280, 7568, 8000, 8288, 8360, 8720, 8828, 8960, 9800, 10340, 10880, 11600, 12320, 12608, 13040, 13328, 13580, 14300, 14768, 14960, 15200, 15488, 16280, 16640, 16928, 17360, 18080, 18368, 18656, 19520, 20240, 21680, 22400, 22688, 23120, 23408, 23840, 24128, 24200, 24920, 25460, 26000, 26720, 27440, 27728, 28160, 29240, 29348, 29600, 29888, 30320, 30800, 31328, 31760, 32480, 33200, 33440, 33488, 34100, 34640, 35288, 35360, 36080, 36800, 37520, 38528, 38960, 39248, 39320, 40040, 41120, 41840, 42560, 42848, 43010, 43280, 44000, 44720, 45008, 45080, 45440, 45920, 46880, 47168, 47600, 47888, 48608, 49280, 49328, 49760, 51128, 51200, 51920, 52640, 53360, 53648, 53900, 54080, 54368, 55088, 55160, 55880, 56240, 56672, 56960, 57200, 57680, 57968, 58400, 58688, 59120, 59840, 60560, 61100, 62000, 62288, 62720, 63008, 63440, 63800, 64400, 64448, 64880, 64988, 65600, 65780, 66608, 67040, 67760, 68480, 68768, 69020, 69200, 69920, 70400, 70928, 71360, 72080, 72800, 73088, 73520, 73808, 74240, 74600, 74960, 75680, 76544, 76868, 77120, 77840, 78128, 78560, 78848, 79640, 80000, 80108, 80900, 81620, 82160, 82880, 83168, 83600, 83888, 84320, 85400, 86048, 86240, 86480, 86768, 87200, 87560, 87920, 88208, 88640, 89180, 90728, 90800, 91520, 91988, 92240, 92960, 93500, 93968, 94688, 94760, 95120, 95480, 96200, 96740, 97280, 98000, 98720, 99008, 99440]

b = 11# = 2310 = 2 * 3 * 5 * 7 * 11
[14, 44, 74, 104, 134, 164, 194, 224, 254, 284, 314, 344, 374, 404, 434, 464, 494, 524, 554, 584, 614, 644, 674, 704, 734, 764, 794, 824, 854, 884, 914, 944, 974, 1004, 1034, 1064, 1094, 1124, 1154, 1184, 1214, 1244, 1274, 1304, 1334, 1364, 1394, 1424, 1454, 1484, 1514, 1544, 1574, 1604, 1634, 1664, 1694, 1724, 1754, 1784, 1814, 1844, 1874, 1904, 1934, 1964, 1994, 2024, 2054, 2084, 2114, 2144, 2174, 2204, 2234, 2264, 2294, 2324, 2354, 2384, 2414, 2444, 2474, 2504, 2534, 2564, 2594, 2624, 2654, 2684, 2714, 2744, 2774, 2804, 2834, 2864, 2894, 2924, 2954, 2984, 3014, 3044, 3074, 3104, 3134, 3164, 3194, 3224, 3254, 3284, 3314, 3344, 3374, 3404, 3434, 3464, 3494, 3524, 3554, 3584, 3614, 3644, 3674, 3704, 3734, 3764, 3794, 3824, 3854, 3884, 3914, 3944, 3974, 4004, 4034, 4064, 4094, 4124, 4154, 4184, 4214, 4244, 4274, 4304, 4334, 4364, 4394, 4400, 4424, 4454, 4484, 4514, 4544, 4574, 4604, 4634, 4664, 4694, 4724, 4754, 4784, 4814, 4844, 4874, 4904, 4934, 4964, 4994, 5024, 5054, 5084, 5114, 5144, 5174, 5204, 5234, 5264, 5294, 5324, 5354, 5384, 5414, 5444, 5474, 5504, 5534, 5564, 5594, 5624, 5654, 5684, 5714, 5744, 5774, 5804, 5834, 5864, 5894, 5924, 5954, 5984, 6014, 6044, 6074, 6104, 6134, 6164, 6194, 6224, 6254, 6284, 6314, 6344, 6374, 6404, 6434, 6464, 6494, 6524, 6554, 6584, 6614, 6644, 6674, 6704, 6734, 6764, 6794, 6824, 6854, 6884, 6914, 6944, 6974, 7004, 7034, 7064, 7094, 7124, 7154, 7184, 7214, 7244, 7274, 7304, 7334, 7364, 7394, 7424, 7454, 7484, 7514, 7544, 7574, 7604, 7634, 7664, 7694, 7724, 7754, 7784, 7814, 7844, 7874, 7904, 7934, 7964, 7994, 8024, 8054, 8084, 8114, 8144, 8174, 8204, 8234, 8264, 8294, 8324, 8354, 8384, 8414, 8444, 8474, 8504, 8534, 8564, 8594, 8624, 8654, 8684, 8714, 8744, 8774, 8804, 8834, 8864, 8894, 8924, 8954, 8984, 9014, 9044, 9074, 9104, 9134, 9164, 9194, 9224, 9254, 9284, 9314, 9344, 9374, 9404, 9434, 9464, 9494, 9524, 9554, 9584, 9614, 9644, 9674, 9704, 9734, 9764, 9794, 9824, 9854, 9884, 9914, 9944, 9974]

b = 13# = 30030 = 2 * 3 * 5 * 7 * 11 * 13
[14, 44, 74, 80, 104, 128, 134, 164, 176, 194, 224, 242, 254, 260, 284, 314, 344, 368, 374, 404, 434, 440, 464, 494, 524, 530, 554, 560, 584, 614, 644, 656, 674, 704, 710, 734, 764, 770, 794, 800, 824, 854, 884, 896, 902, 914, 944, 962, 974, 980, 998, 1004, 1034, 1040, 1064, 1088, 1094, 1124, 1154, 1184, 1214, 1232, 1244, 1250, 1274, 1298, 1304, 1334, 1364, 1376, 1382, 1394, 1424, 1430, 1454, 1484, 1514, 1520, 1544, 1568, 1574, 1604, 1634, 1664, 1682, 1694, 1712, 1724, 1754, 1760, 1778, 1784, 1814, 1820, 1826, 1844, 1874, 1904, 1910, 1934, 1958, 1964, 1994, 2000, 2006, 2024, 2054, 2084, 2096, 2114, 2144, 2174, 2204, 2234, 2240, 2264, 2288, 2294, 2324, 2354, 2384, 2414, 2420, 2438, 2444, 2474, 2486, 2504, 2528, 2534, 2564, 2576, 2594, 2624, 2654, 2684, 2714, 2726, 2744, 2768, 2774, 2804, 2816, 2834, 2864, 2870, 2882, 2894, 2912, 2924, 2954, 2960, 2966, 2984, 3014, 3044, 3074, 3080, 3104, 3134, 3158, 3164, 3194, 3224, 3248, 3254, 3266, 3284, 3296, 3314, 3344, 3362, 3374, 3392, 3404, 3410, 3434, 3464, 3494, 3500, 3524, 3536, 3542, 3554, 3584, 3608, 3614, 3644, 3674, 3680, 3704, 3734, 3764, 3794, 3824, 3854, 3872, 3884, 3890, 3914, 3920, 3938, 3944, 3974, 4004, 4022, 4034, 4064, 4070, 4094, 4124, 4130, 4154, 4160, 4184, 4214, 4220, 4244, 4256, 4274, 4304, 4334, 4352, 4364, 4394, 4400, 4418, 4424, 4454, 4466, 4484, 4508, 4514, 4544, 4550, 4562, 4574, 4592, 4598, 4604, 4634, 4640, 4664, 4694, 4724, 4754, 4760, 4784, 4814, 4844, 4874, 4880, 4904, 4928, 4934, 4964, 4994, 5012, 5024, 5054, 5078, 5084, 5114, 5120, 5126, 5138, 5144, 5174, 5204, 5234, 5264, 5294, 5324, 5354, 5384, 5408, 5414, 5444, 5456, 5474, 5504, 5522, 5534, 5564, 5594, 5600, 5606, 5624, 5654, 5684, 5714, 5720, 5744, 5750, 5774, 5804, 5834, 5840, 5858, 5864, 5894, 5924, 5936, 5954, 5984, 6002, 6014, 6020, 6032, 6044, 6074, 6104, 6134, 6164, 6182, 6194, 6200, 6224, 6254, 6272, 6284, 6314, 6344, 6374, 6380, 6404, 6434, 6464, 6Tables not complete presented here, because too long494, 6512, 6524, 6530, 6554, 6560, 6578, 6584, 6608, 6614, 6644, 6662, 6674, 6704, 6710, 6734, 6764, 6776, 6794, 6800, 6824, 6854, 6860, 6866, 6884, 6902, 6914, 6944, 6974, 6992, 7004, 7034, 7040, 7058, 7064, 7094, 7106, 7124, 7154, 7184, 7190, 7214, 7238, 7244, 7274, 7280, 7298, 7304, 7334, 7364, 7394, 7424, 7454, 7484, 7514, 7520, 7544, 7568, 7574, 7586, 7604, 7616, 7634, 7664, 7694, 7700, 7724, 7754, 7766, 7784, 7814, 7844, 7874, 7904, 7910, 7934, 7952, 7964, 7994, 8000, 8018, 8024, 8048, 8054, 8084, 8096, 8114, 8144, 8162, 8174, 8180, 8204, 8234, 8246, 8264, 8288, 8294, 8324, 8354, 8360, 8384, 8414, 8432, 8444, 8450, 8474, 8504, 8534, 8540, 8564, 8576, 8594, 8624, 8642, 8654, 8684, 8690, 8714, 8720, 8744, 8774, 8804, 8834, 8840, 8864, 8894, 8924, 8954, 8960, 8984, 9014, 9044, 9062, 9074, 9104, 9134, 9152, 9164, 9170, 9194, 9200, 9218, 9224, 9254, 9284, 9296, 9302, 9314, 9344, 9350, 9374, 9404, 9434, 9440, 9464, 9494, 9524, 9548, 9554, 9584, 9614, 9632, 9644, 9674, 9680, 9698, 9704, 9728, 9734, 9746, 9764, 9794, 9800, 9824, 9830, 9854, 9884, 9890, 9914, 9944, 9968, 9974]

b = 17# = 510510 = 2 * 3 * 5 * 7 * 11 * 13 * 17
[14, 44, 74, 104, 134, 164, 194, 224, 254, 284, 314, 344, 374, 404, 434, 440, 464, 494, 524, 554, 560, 584, 614, 638, 644, 674, 704, 734, 764, 782, 794, 824, 854, 884, 914, 944, 974, 980, 1004, 1034, 1064, 1094, 1124, 1154, 1184, 1214, 1232, 1244, 1274, 1304, 1334, 1364, 1394, 1424, 1430, 1454, 1484, 1514, 1544, 1574, 1604, 1634, 1664, 1694, 1724, 1754, 1760, 1784, 1814, 1844, 1874, 1904, 1934, 1964, 1994, 2024, 2054, 2084, 2114, 2144, 2174, 2204, 2234, 2240, 2264, 2288, 2294, 2324, 2354, 2384, 2414, 2444, 2474, 2504, 2534, 2564, 2594, 2600, 2624, 2654, 2684, 2714, 2744, 2774, 2804, 2816, 2834, 2864, 2894, 2924, 2954, 2960, 2984, 3014, 3044, 3074, 3080, 3104, 3134, 3164, 3194, 3224, 3248, 3254, 3284, 3314, 3344, 3374, 3404, 3434, 3464, 3494, 3500, 3524, 3554, 3584, 3614, 3644, 3674, 3680, 3704, 3734, 3752, 3764, 3794, 3824, 3854, 3884, 3914, 3920, 3944, 3950, 3974, 4004, 4034, 4064, 4094, 4124, 4154, 4184, 4214, 4244, 4274, 4304, 4334, 4346, 4364, 4394, 4400, 4424, 4454, 4484, 4508, 4514, 4544, 4574, 4604, 4634, 4664, 4694, 4724, 4754, 4760, 4784, 4814, 4844, 4874, 4904, 4928, 4934, 4940, 4964, 4994, 5024, 5054, 5084, 5114, 5144, 5174, 5204, 5234, 5264, 5294, 5324, 5336, 5354, 5384, 5390, 5414, 5444, 5474, 5504, 5534, 5564, 5588, 5594, 5600, 5624, 5654, 5684, 5714, 5720, 5744, 5774, 5804, 5834, 5864, 5894, 5924, 5936, 5954, 5984, 6014, 6044, 6074, 6104, 6134, 6164, 6194, 6224, 6254, 6284, 6314, 6344, 6374, 6404, 6434, 6440, 6464, 6494, 6524, 6554, 6578, 6584, 6614, 6644, 6674, 6704, 6734, 6764, 6776, 6794, 6824, 6854, 6884, 6914, 6920, 6944, 6974, 7004, 7034, 7040, 7064, 7094, 7124, 7154, 7184, 7214, 7244, 7274, 7280, 7304, 7334, 7364, 7394, 7424, 7454, 7484, 7514, 7544, 7568, 7574, 7604, 7634, 7664, 7694, 7700, 7724, 7754, 7784, 7814, 7844, 7874, 7904, 7910, 7934, 7964, 7994, 8024, 8054, 8084, 8114, 8120, 8144, 8174, 8204, 8234, 8264, 8288, 8294, 8324, 8354, 8360, 8372, 8384, 8414, 8444, 8474, 8504, 8534, 8564, 8594, 8624, 8654, 8684, 8714, 8744, 8774, 8804, 8834, 8864, 8894, 8924, 8954, 8960, 8984, 8990, 9014, 9044, 9074, 9104, 9134, 9152, 9164, 9194, 9224, 9254, 9284, 9296, 9314, 9344, 9350, 9374, 9404, 9434, 9464, 9494, 9524, 9548, 9554, 9584, 9614, 9644, 9674, 9680, 9704, 9734, 9764, 9794, 9800, 9824, 9854, 9884, 9890, 9914, 9944, 9974]

b = 19# = 9699690 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19
[14, 44, 56, 62, 74, 80, 92, 98, 104, 110, 116, 128, 134, 140, 164, 176, 194, 200, 206, 224, 230, 242, 254, 260, 284, 308, 314, 320, 344, 350, 356, 362, 368, 374, 392, 404, 416, 422, 434, 440, 464, 476, 494, 506, 512, 518, 524, 530, 554, 560, 572, 584, 614, 638, 644, 650, 656, 662, 668, 674, 680, 704, 728, 734, 764, 770, 782, 794, 800, 812, 818, 824, 830, 836, 854, 884, 890, 896, 902, 914, 920, 944, 962, 968, 974, 980, 986, 998, 1004, 1034, 1040, 1058, 1064, 1088, 1094, 1100, 1124, 1142, 1148, 1154, 1160, 1166, 1184, 1196, 1214, 1232, 1244, 1274, 1280, 1292, 1298, 1304, 1316, 1322, 1334, 1358, 1364, 1376, 1394, 1400, 1424, 1430, 1448, 1454, 1472, 1484, 1496, 1508, 1514, 1520, 1544, 1562, 1568, 1574, 1604, 1610, 1628, 1634, 1640, 1652, 1664, 1670, 1682, 1694, 1700, 1724, 1736, 1748, 1754, 1760, 1766, 1784, 1814, 1820, 1826, 1844, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1904, 1916, 1934, 1952, 1958, 1964, 1988, 1994, 2000, 2006, 2024, 2030, 2042, 2054, 2072, 2078, 2084, 2090, 2114, 2120, 2132, 2144, 2156, 2162, 2174, 2204, 2222, 2228, 2234, 2240, 2264, 2288, 2294, 2300, 2324, 2330, 2354, 2360, 2378, 2384, 2390, 2408, 2414, 2420, 2438, 2444, 2450, 2456, 2474, 2480, 2486, 2492, 2504, 2528, 2534, 2540, 2546, 2552, 2564, 2576, 2594, 2600, 2618, 2624, 2654, 2660, 2684, 2714, 2720, 2726, 2744, 2750, 2756, 2762, 2774, 2804, 2816, 2828, 2834, 2840, 2852, 2864, 2870, 2882, 2894, 2900, 2912, 2918, 2924, 2948, 2954, 2960, 2984, 2990, 2996, 3008, 3014, 3044, 3068, 3074, 3080, 3086, 3104, 3128, 3134, 3146, 3164, 3170, 3194, 3200, 3212, 3224, 3230, 3248, 3254, 3266, 3278, 3284, 3314, 3320, 3326, 3332, 3338, 3344, 3362, 3374, 3380, 3386, 3392, 3404, 3410, 3416, 3434, 3440, 3464, 3476, 3482, 3494, 3500, 3524, 3536, 3542, 3554, 3560, 3584, 3608, 3614, 3632, 3644, 3662, 3668, 3674, 3680, 3698, 3704, 3710, 3734, 3740, 3752, 3764, 3770, 3782, 3788, 3794, 3800, 3806, 3818, 3824, 3836, 3848, 3854, 3860, 3872, 3878, 3884, 3914, 3920, 3938, 3944, 3950, 3956, 3968, 3974, 4004, 4010, 4022, 4034, 4040, 4064, 4070, 4088, 4094, 4100, 4124, 4130, 4136, 4154, 4160, 4166, 4172, 4184, 4202, 4214, 4220, 4232, 4244, 4256, 4262, 4268, 4274, 4280, 4292, 4304, 4310, 4316, 4334, 4340, 4346, 4352, 4364, 4370, 4394, 4400, 4418, 4424, 4454, 4466, 4472, 4478, 4484, 4508, 4514, 4520, 4532, 4544, 4550, 4562, 4574, 4592, 4598, 4604, 4628, 4634, 4640, 4646, 4652, 4664, 4676, 4694, 4724, 4730, 4754, 4760, 4784, 4796, 4814, 4832, 4844, 4862, 4874, 4880, 4886, 4904, 4922, 4928, 4934, 4940, 4946, 4964, 4988, 4994, 5000]

b = 23# = 223092870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23
[14, 32, 44, 56, 62, 74, 80, 92, 98, 104, 116, 122, 134, 140, 164, 176, 194, 200, 212, 224, 230, 242, 254, 260, 284, 296, 308, 314, 320, 332, 338, 344, 350, 356, 362, 368, 374, 392, 398, 404, 416, 428, 434, 440, 452, 464, 476, 494, 506, 512, 524, 530, 536, 554, 560, 572, 584, 614, 626, 638, 644, 656, 668, 674, 680, 692, 704, 728, 734, 752, 758, 764, 770, 782, 794, 800, 812, 824, 836, 854, 884, 890, 896, 902, 914, 920, 926, 932, 944, 950, 956, 968, 974, 980, 1004, 1022, 1028, 1034, 1040, 1052, 1058, 1064, 1088, 1094, 1100, 1124, 1136, 1148, 1154, 1160, 1166, 1172, 1184, 1190, 1196, 1214, 1220, 1226, 1232, 1244, 1250, 1274, 1280, 1286, 1292, 1304, 1316, 1334, 1340, 1352, 1358, 1364, 1394, 1400, 1412, 1418, 1424, 1430, 1442, 1448, 1454, 1472, 1484, 1496, 1502, 1508, 1514, 1520, 1532, 1544, 1550, 1568, 1574, 1580, 1604, 1610, 1616, 1628, 1634, 1640, 1652, 1664, 1682, 1694, 1700, 1718, 1724, 1736, 1748, 1754, 1760, 1772, 1778, 1784, 1790, 1814, 1820, 1826, 1844, 1856, 1874, 1880, 1886, 1892, 1904, 1916, 1934, 1946, 1958, 1964, 1976, 1988, 1994, 2000, 2012, 2024, 2030, 2054, 2072, 2078, 2084, 2090, 2114, 2120, 2132, 2144, 2156, 2162, 2174, 2192, 2204, 2210, 2222, 2228, 2234, 2240, 2252, 2264, 2270, 2276, 2288, 2294, 2300, 2312, 2324, 2330, 2342, 2354, 2360, 2366, 2372, 2378, 2384, 2408, 2414, 2420, 2438, 2444, 2450, 2456, 2468, 2474, 2480, 2486, 2492, 2504, 2534, 2540, 2546, 2552, 2564, 2576, 2594, 2600, 2606, 2612, 2618, 2624, 2654, 2660, 2672, 2684, 2690, 2696, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2774, 2804, 2816, 2822, 2828, 2834, 2840, 2852, 2864, 2870, 2882, 2894, 2900, 2912, 2924, 2936, 2942, 2948, 2954, 2960, 2972, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3038, 3044, 3068, 3074, 3080, 3092, 3098, 3104, 3128, 3134, 3158, 3164, 3194, 3200, 3212, 3224, 3230, 3236, 3248, 3254, 3266, 3284, 3296, 3314, 3320, 3332, 3344, 3374, 3380, 3392, 3398, 3404, 3410, 3416, 3422, 3434, 3440, 3452, 3464, 3476, 3494, 3500, 3512, 3518, 3524, 3530, 3536, 3542, 3554, 3560, 3572, 3584, 3596, 3608, 3614, 3632, 3644, 3650, 3662, 3668, 3674, 3680, 3692, 3698, 3704, 3710, 3728, 3734, 3740, 3752, 3764, 3770, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3836, 3848, 3854, 3860, 3872, 3878, 3884, 3914, 3920, 3926, 3932, 3938, 3944, 3956, 3974, 3986, 3992, 4004, 4034, 4040, 4052, 4058, 4064, 4088, 4094, 4100, 4124, 4130, 4136, 4154, 4160, 4172, 4184, 4190, 4202, 4214, 4232, 4244, 4256, 4262, 4268, 4274, 4280, 4292, 4304, 4316, 4322, 4328, 4334, 4340, 4346, 4364, 4370, 4382, 4388, 4394, 4400, 4412, 4424, 4454, 4466, 4472, 4478, 4484, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4562, 4568, 4574, 4586, 4592, 4598, 4604, 4616, 4628, 4634, 4640, 4646, 4652, 4664, 4670, 4676, 4694, 4718, 4724, 4730, 4754, 4760, 4772, 4784, 4796, 4814, 4844, 4850, 4862, 4874, 4880, 4892, 4904, 4916, 4922, 4928, 4934, 4940, 4952, 4958, 4964, 4982, 4988, 4994, 5000]

b = 29# = 6469693230 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29
[14, 20, 26, 32, 44, 56, 62, 68, 74, 80, 92, 98, 104, 110, 116, 122, 134, 140, 146, 152, 164, 176, 188, 194, 200, 212, 224, 230, 242, 248, 254, 260, 272, 284, 290, 296, 314, 320, 332, 344, 350, 356, 362, 368, 374, 386, 392, 398, 404, 416, 428, 434, 440, 464, 476, 482, 494, 500, 506, 512, 524, 530, 536, 554, 560, 566, 572, 584, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 674, 680, 692, 704, 710, 716, 722, 728, 734, 752, 758, 764, 770, 776, 782, 788, 794, 800, 812, 818, 824, 830, 836, 842, 848, 854, 860, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 956, 962, 968, 974, 980, 986, 1004, 1016, 1022, 1028, 1034, 1040, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1112, 1124, 1148, 1154, 1160, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1232, 1238, 1244, 1250, 1256, 1274, 1280, 1286, 1292, 1304, 1322, 1328, 1334, 1340, 1352, 1358, 1364, 1376, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1472, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1532, 1538, 1544, 1550, 1562, 1568, 1574, 1580, 1592, 1604, 1610, 1616, 1628, 1634, 1640, 1652, 1658, 1664, 1682, 1688, 1694, 1700, 1724, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1844, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1904, 1910, 1922, 1934, 1940, 1946, 1952, 1964, 1976, 1988, 1994, 2000, 2012, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2072, 2078, 2084, 2096, 2114, 2120, 2132, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2192, 2204, 2210, 2228, 2234, 2240, 2246, 2258, 2264, 2276, 2282, 2288, 2294, 2300, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2372, 2378, 2384, 2402, 2408, 2414, 2420, 2438, 2444, 2450, 2456, 2474, 2480, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2552, 2564, 2576, 2582, 2588, 2594, 2600, 2606, 2618, 2624, 2636, 2642, 2654, 2660, 2666, 2672, 2684, 2690, 2696, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2792, 2804, 2816, 2834, 2840, 2852, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2912, 2918, 2924, 2936, 2948, 2954, 2960, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3032, 3044, 3050, 3056, 3068, 3074, 3080, 3086, 3092, 3104, 3116, 3122, 3128, 3134, 3140, 3146, 3158, 3164, 3170, 3194, 3200, 3212, 3224, 3230, 3236, 3242, 3248, 3254, 3266, 3272, 3284, 3290, 3296, 3308, 3314, 3320, 3332, 3338, 3344, 3356, 3362, 3374, 3380, 3392, 3398, 3404, 3410, 3416, 3422, 3434, 3440, 3452, 3464, 3482, 3488, 3494, 3500, 3506, 3512, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3584, 3590, 3596, 3602, 3608, 3614, 3632, 3644, 3662, 3668, 3674, 3680, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3752, 3758, 3764, 3770, 3776, 3794, 3800, 3806, 3812, 3818, 3824, 3836, 3842, 3848, 3854, 3860, 3872, 3884, 3896, 3914, 3920, 3926, 3944, 3956, 3962, 3968, 3974, 3992, 4004, 4010, 4022, 4028, 4034, 4040, 4052, 4058, 4064, 4076, 4082, 4088, 4094, 4100, 4112, 4118, 4124, 4130, 4136, 4148, 4154, 4160, 4172, 4178, 4184, 4190, 4202, 4208, 4214, 4220, 4232, 4238, 4244, 4250, 4256, 4262, 4274, 4280, 4292, 4304, 4310, 4316, 4322, 4334, 4340, 4346, 4352, 4364, 4370, 4376, 4388, 4394, 4400, 4424, 4430, 4436, 4442, 4454, 4460, 4466, 4472, 4484, 4496, 4508, 4514, 4520, 4526, 4532, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4616, 4628, 4634, 4640, 4646, 4652, 4664, 4676, 4682, 4688, 4694, 4712, 4718, 4724, 4748, 4754, 4760, 4766, 4778, 4784, 4796, 4802, 4808, 4814, 4820, 4832, 4844, 4850, 4856, 4874, 4880, 4886, 4892, 4904, 4916, 4922, 4928, 4934, 4940, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 31# = 200560490130 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31
[2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 37# = 7420738134810 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37
[8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 41# = 304250263527210 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41
[8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 43# = 13082761331670030 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43
[2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 47# = 614889782588491410 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47
[2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

...
...

b = 97# = 100# = 2305567963945518424753102147331756070
[2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 454, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2274, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3184]

...
...

b = 997# = 1000# = 19590340644999083431262508198206381046123972390589368223882605328968666316379870661851951648789482321596229559115436019149189529725215266728292282990852649023362731392404017939142010958261393634959471483757196721672243410067118516227661133135192488848989914892157188308679896875137439519338903968094905549750386407106033836586660683539201011635917900039904495065203299749542985993134669814805318474080581207891125910
[2, 8, 14, 20, 26, 32, 34, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 174, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 244, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 384, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 454, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 594, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 664, 668, 674, 680, 686, 692, 698, 704, 710, 714, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 804, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 874, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1014, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1084, 1328, 1504, 1748, 1924]
In this table, for all primorial bases b = p# for p (prime) from 3 to 47, one can see all exponents e ranked in order, up to a certain limit, such that s(b^e) is abundant.
At the end of the table, we have added two very large primorial bases : b = 97# and b = 997#.


Some explanations, remarks and a curious question (in bold) :

Let's describe what we observe, starting at the bottom of the table.

For b = 97# and b = 997#.
We see that all exponents e = 2 + 6 * k, (k integer) are present.
There are even other exponents that are not of the form 2 + 6 * k that appear in addition.
For example, for b = 97#, we have three extra exponents which are : 454, 2274 and 3184.
For b = 997#, we have a larger list : 34, 174, 244, 384, 454, 594, 664, 714, 804, 874, 1014, 1504, 1924.
But I was totally unable to formulate a conjecture that says what these added numbers are !

For b = 7# to b = 47#.
All the exponents we see in the table are of the form e = 2 + 6 * k, without exception.
But I think this would not be true if we were considering very large exponents with very large k. Exponents of some other form would most likely be added.
On the other hand, we can note that only from b = 43# onwards all exponents of the form 2 + 6 * k appear in the table, hence the formulation of the conjecture (140) which specifies "(p prime >41)".
Let us see for which bases some exponents of the form 2 + 6 * k are missing and for which we see them all :
For b = 47#, all exponents are present.
For b = 43#, all exponents are present.
For b = 41#, the exponent e = 2 is missing (k = 0)
For b = 37#, the exponent e = 2 is missing (k = 0)
For b = 31#, all exponents are present.
For b = 29#, many exponents are missing : e = 2 (k = 0), e = 8 (k = 1), e = 38 (k = 6), e = 50 (k = 8), ...
For b = 23#, even more exponents are missing : e = 2 (k = 0), e = 8 (k = 1), e = 20 (k = 3), ...
And there are always more exponents of the form 2 + 6 * k missing for b = p# when p decreases to 7.
...
...
For b = 7#, almost all exponents of the form e = 2 + 6 * k are missing and there, it is the opposite, there are only very few present : e = 80 (k = 13), e = 440 (k = 73), e = 1520 (k = 253), e = 1700 (k = 283), ...

For b = 3# to b = 5#.
The primorial numbers 6 = 3# and 30 = 5# really behave very differently from the other primorial numbers !!!
It's just the opposite. None of the exponents present in the table are of the form e = 2 + 6 * k and this is very curious !
Why does everything switch when we go from the base b = 5# to the base 7# ?
I did not succeed in stating a conjecture which allows to predict what are the exponents e present in the table for these two very special bases !
For the base b = 3#, there seem to be exponents of the form e = 720 * k and e = 1080 * k, k integer, but, I did not check this.

For b = 2#.
Refer to the conjecture (137).
garambois is offline   Reply With Quote