Thursday, July 28, 2022

Cycles in Éric Angelini's sum-and-erase

Éric's article on sum-and-erase is here. I wanted a place where I could post known cycles, so I will do that in this listing [index.  name = c&start-string  (cycle length)  {precursors}]:

0.  c0  (1)  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 00, 01, 02, 03, 05, 06, 07, 08, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, ...}

1.  c3374  (583792)  {04, 09, 25, 31, 63, 67, 69, 77, 92, 96, 99, 023, 027, 031, 040, 042, 044, 049, 067, 074, 075, 081, 082, 090, 093, 099, 105, 109, 116, 133, 138, 148, 152, 161, 162, 163, 168, 174, 194, 197, 198, 206, 216, 253, 257, 258, 285, 307, 314, 327, 352, 370, 382, 389, 398, 422, 423, 430, 463, 477, 482, 483, 496, 503, 521, 543, 544, 568, 581, 582, 590, 592, 599, 604, 605, 610, 613, 630, 635, 639, 645, 656, 665, 667, 668, 676, 681, 682, 685, 686, 688, 708, 722, 736, 737, 742, 758, 766, 773, 782, 801, 809, 810, 814, 821, 839, 841, 848, 849, 856, 858, 867, 875, 884, 885, 894, 902, 906, 923, 932, 944, 958, 977, 982, 988, 0119, 0135, 0144, 0147, 0152, 0163, 0166, 0176, 0191, 0199, 0203, 0207, 0216, 0230, 0235, 0244, 0253, 0256, 0259, 0267, 0270, 0275, 0279, 0299, 0301, 0310, 0314, 0317, 0333, 0349, 0361, 0375, 0378, 0385, 0387, 0400, 0402, 0404, 0409, 0420, 0426, 0429, 0440, 0448, 0482, 0484, 0486, 0490, 0498, 0521, 0539, 0559, 0562, 0578, 0592, 0595, 0607, 0619, 0645, 0656, 0658, 0665, 0668, 0670, 0671, 0673, 0686, 0698, 0704, 0705, 0713, 0716, 0735, 0736, 0738, 0740, 0750, 0767, 0768, 0776, 0794, 0801, 0802, 0810, 0813, 0814, 0817, 0819, 0820, 0824, 0829, 0831, 0847, 0855, 0856, 0866, 0867, 0900, 0903, 0909, 0911, 0916, 0919, 0925, 0929, 0930, 0938, 0957, 0958, 0982, 0990, 0991, 0992, 0999, 1005, 1009, 1016, 1019, 1027, 1050, 1056, 1059, 1061, 1062, 1082, 1090, 1091, 1099, 1106, 1109, 1119, 1120, 1163, 1168, 1191, 1192, 1201, 1210, 1222, 1238, 1239, 1250, 1253, 1254, 1264, 1268, 1277, 1285, 1289, 1299, 1307, 1327, 1333, 1335, 1337, 1339, 1352, 1353, 1364, 1365, 1370, 1373, 1376, 1378, 1393, 1396, 1422, 1495, 1520, 1526, 1528, 1530, 1545, 1554, 1557, 1559, 1566, 1575, 1578, 1593, 1595, 1604, 1613, 1618, 1628, 1629, 1630, 1631, 1637, 1648, 1649, 1668, 1679, 1681, 1685, 1686, 1724, 1728, 1729, 1739, 1754, 1760, 1762, 1767, 1776, 1792, 1793, 1794, 1823, 1825, 1832, 1840, 1843, 1853, 1866, 1867, 1894, 1904, 1907, 1908, 1911, 1912, 1921, 1923, 1929, 1933, 1934, 1938, 1970, 1987, 1992, 2006, 2019, 2028, 2047, 2048, 2053, 2058, 2060, 2068, 2074, 2082, 2085, 2091, 2094, 2099, 2109, 2126, 2133, 2134, 2157, 2162, 2163, 2168, 2169, 2173, 2188, 2194, 2208, 2216, 2222, 2230, 2234, 2249, 2254, 2264, 2280, 2291, 2295, 2299, 2302, 2307, 2314, 2320, 2324, 2342, 2344, 2347, 2348, 2351, 2357, 2358, 2361, 2370, 2384, 2389, 2391, 2394, 2395, 2398, 2413, 2414, 2416, 2418, 2429, 2441, 2458, 2471, 2480, 2481, 2492, 2503, 2524, 2531, 2542, 2545, 2554, 2556, 2565, 2576, 2584, 2604, 2624, 2642, 2643, 2644, 2655, 2668, 2679, 2686, 2687, 2704, 2740, 2757, 2758, 2763, 2764, 2775, 2777, 2784, 2786, 2793, 2802, 2805, 2818, 2820, 2837, 2839, 2843, 2848, 2850, 2856, 2875, 2881, 2884, 2897, 2904, 2906, 2912, 2917, 2918, 2921, 2925, 2929, 2948, 2951, 2952, 2964, 2992, 3007, 3019, 3033, 3054, 3057, 3061, 3070, 3071, 3089, 3091, 3095, 3098, 3109, 3117, 3125, 3129, 3136, 3140, 3146, 3148, 3163, 3164, 3167, 3171, 3176, 3177, 3199, 3213, 3218, 3231, 3237, 3244, 3254, 3256, 3273, 3287, 3298, 3299, 3303, 3316, 3321, 3327, 3330, 3335, 3352, 3353, 3371, 3374, 3387, 3396, 3398, 3457, 3462, 3476, 3502, 3509, 3512, 3515, 3523, 3528, 3532, 3533, 3546, 3551, 3564, 3578, 3604, 3611, 3612, 3619, 3644, 3654, 3655, 3668, 3675, 3686, 3687, 3700, 3701, 3707, 3709, 3713, 3724, 3731, 3734, 3743, 3747, 3757, 3767, 3770, 3774, 3775, 3776, 3789, 3792, 3794, 3804, 3809, 3812, 3814, 3821, 3837, 3851, 3855, 3856, 3858, 3859, 3866, 3872, 3873, 3885, 3888, 3896, 3905, 3911, 3922, 3929, 3936, 3938, 3942, 3950, 3963, 3967, 3969, 3972, 3983, 3986, 3989, 3992, 3996, 3998, 4019, 4023, 4035, 4053, 4063, 4069, 4091, 4096, 4097, 4099, 4102, 4109, 4123, 4139, 4146, 4155, 4163, 4164, 4176, 4196, 4202, 4218, 4220, 4228, 4235, 4239, 4253, 4282, 4283, 4299, 4300, 4301, 4307, 4347, 4348, 4349, 4361, 4368, 4370, 4371, 4374, 4376, 4378, 4380, 4381, 4384, 4387, 4394, 4399, 4416, 4437, 4438, 4439, 4452, 4458, 4461, 4483, 4491, 4492, 4506, 4516, 4524, 4527, 4531, 4542, 4548, 4578, 4584, 4598, 4612, 4614, 4616, 4625, 4627, 4628, 4641, 4661, 4668, 4671, 4686, 4698, 4701, 4710, 4731, 4767, 4776, 4793, 4796, 4815, 4826, 4834, 4843, 4850, 4866, 4888, 4890, 4911, 4914, 4922, 4924, 4929, 4935, 4938, 4939, 4941, 4942, 4959, 4960, 4970, 4992, 4993, 4995, 5003, 5019, 5030, 5031, 5033, 5034, 5035, 5038, 5039, 5053, 5063, 5067, 5088, 5091, 5109, 5113, 5131, 5134, 5142, 5163, 5166, 5176, 5186, 5196, 5198, 5218, 5236, 5244, 5249, 5263, 5264, 5274, 5286, 5296, 5299, 5307, 5333, 5339, 5361, 5366, 5370, 5371, 5384, 5388, 5391, 5393, 5428, 5429, 5439, 5446, 5448, 5455, 5456, 5464, 5465, 5483, 5484, 5487, 5493, 5496, 5503, 5545, 5546, 5554, 5555, 5562, 5566, 5578, 5582, 5584, 5604, 5616, 5625, 5630, 5637, 5648, 5652, 5656, 5661, 5665, 5668, 5686, 5694, 5720, 5742, 5758, 5760, 5761, 5763, 5767, 5776, 5779, 5781, 5782, 5785, 5794, 5797, 5810, 5814, 5825, 5827, 5832, 5845, 5852, 5854, 5863, 5866, 5870, 5871, 5900, 5929, 5930, 5937, 5938, 5964, 5979, 5982, 5992, 5997, 6004, 6005, 6014, 6019, 6021, 6040, 6041, 6050, 6056, 6065, 6078, 6091, 6099, 6100, 6104, 6106, 6107, 6109, 6130, 6134, 6155, 6160, 6174, 6177, 6225, 6235, 6236, 6237, 6244, 6252, 6253, 6263, 6268, 6277, 6286, 6287, 6288, 6298, 6299, 6300, 6307, 6309, 6318, 6321, 6351, 6370, 6372, 6377, 6379, 6404, 6405, 6416, 6417, 6429, 6430, 6440, 6453, 6461, 6473, 6485, 6489, 6490, 6499, 6545, 6554, 6559, 6569, 6572, 6578, 6588, 6595, 6596, 6605, 6610, 6623, 6628, 6641, 6659, 6680, 6682, 6691, 6698, 6705, 6713, 6714, 6718, 6723, 6743, 6753, 6781, 6794, 6803, 6804, 6806, 6812, 6823, 6826, 6830, 6834, 6837, 6852, 6854, 6860, 6862, 6873, 6894, 6899, 6904, 6913, 6915, 6916, 6922, 6923, 6929, 6938, 6942, 6949, 6961, 6968, 6975, 6986, 6992, 6994, 6999, 7008, 7019, 7035, 7037, 7056, 7067, 7073, 7076, 7078, 7080, 7084, 7087, 7091, 7099, 7102, 7109, 7147, 7150, 7158, 7163, 7174, 7178, 7187, 7195, 7205, 7227, 7230, 7235, 7240, 7244, 7253, 7260, 7263, 7272, 7279, 7288, 7294, 7297, 7299, 7305, 7318, 7340, 7358, 7361, 7365, 7366, 7377, 7385, 7458, 7490, 7493, 7495, 7499, 7502, 7508, 7516, 7517, 7526, 7529, 7530, 7542, 7547, 7561, 7563, 7571, 7574, 7578, 7581, 7587, 7604, 7611, 7614, 7620, 7621, 7626, 7627, 7635, 7655, 7657, 7662, 7668, 7672, 7675, 7680, 7682, 7686, 7703, 7706, 7708, 7714, 7718, 7722, 7729, 7737, 7751, 7754, 7758, 7762, 7765, 7773, 7798, 7824, 7825, 7833, 7841, 7849, 7860, 7863, 7866, 7892, 7904, 7909, 7912, 7920, 7929, 7931, 7936, 7938, 7940, 7941, 7942, 7945, 7950, 7958, 7978, 7987, 7990, 7992, 7999, 8001, 8009, 8010, 8012, 8013, 8017, 8019, 8024, 8029, 8031, 8041, 8047, 8053, 8067, 8090, 8091, 8100, 8108, 8109, 8129, 8143, 8152, 8156, 8160, 8162, 8163, 8173, 8179, 8180, 8204, 8224, 8229, 8234, 8235, 8239, 8242, 8244, 8253, 8256, 8279, 8292, 8299, 8307, 8313, 8323, 8324, 8331, 8332, 8341, 8344, 8358, 8361, 8369, 8370, 8374, 8385, 8391, 8401, 8402, 8403, 8407, 8408, 8422, 8436, 8446, 8452, 8456, 8464, 8480, 8492, 8506, 8517, 8518, 8527, 8531, 8534, 8537, 8561, 8562, 8566, 8581, 8604, 8626, 8633, 8644, 8647, 8657, 8662, 8669, 8672, 8679, 8692, 8696, 8707, 8716, 8723, 8724, 8726, 8727, 8731, 8736, 8742, 8765, 8767, 8770, 8772, 8776, 8788, 8791, 8794, 8810, 8835, 8840, 8851, 8878, 8887, 8891, 8893, 8905, 8918, 8929, 8938, 8944, 8952, 8961, 8976, 8981, 8983, 8992, 9002, 9006, 9017, 9020, 9025, 9029, 9038, 9044, 9058, 9060, 9077, 9088, 9092, 9104, 9105, 9128, 9143, 9144, 9149, 9163, 9178, 9180, 9182, 9189, 9194, 9198, 9211, 9227, 9229, 9235, 9244, 9247, 9253, 9272, 9284, 9288, 9292, 9302, 9307, 9336, 9347, 9361, 9362, 9363, 9370, 9377, 9379, 9382, 9389, 9397, 9398, 9407, 9416, 9435, 9444, 9462, 9468, 9475, 9478, 9483, 9508, 9516, 9518, 9556, 9565, 9567, 9568, 9578, 9583, 9604, 9615, 9617, 9618, 9626, 9633, 9641, 9645, 9658, 9662, 9668, 9686, 9688, 9689, 9698, 9704, 9707, 9709, 9710, 9749, 9750, 9761, 9767, 9770, 9776, 9790, 9794, 9801, 9866, 9870, 9902, 9914, 9918, 9922, 9937, 9938, 9968, 9970, 9974}

2.  c083433  (120621)  {092, 245, 351, 395, 413, 433, 468, 571, 802, 0382, 0479, 0588, 0594, 0691, 0848, 0884, 0902, 0920, 1038, 1142, 1170, 1369, 1380, 1412, 1421, 1433, 1479, 1482, 1560, 1584, 1645, 1666, 1701, 1710, 1802, 1848, 1884, 1986, 2096, 2148, 2235, 2251, 2325, 2352, 2450, 2477, 2512, 2521, 2555, 2601, 2634, 2649, 2688, 2761, 2955, 2970, 2999, 3358, 3479, 3510, 3519, 3538, 3544, 3583, 3699, 3702, 3769, 3779, 3797, 3802, 3848, 3884, 3917, 3949, 3981, 3994, 4012, 4103, 4112, 4121, 4122, 4138, 4160, 4165, 4183, 4286, 4650, 4707, 4770, 4790, 4792, 4802, 4825, 4870, 4909, 4933, 4990, 5078, 5108, 5277, 5427, 5479, 5602, 5786, 5802, 5808, 5823, 5848, 5880, 5884, 6335, 6353, 6385, 6411, 6456, 6465, 6479, 6491, 6591, 6645, 6669, 6694, 6696, 6802, 6848, 6884, 6946, 6964, 6966, 7014, 7098, 7228, 7282, 7291, 7444, 7579, 7588, 7597, 7759, 7802, 7816, 7848, 7884, 7903, 7946, 8002, 8020, 8021, 8288, 8376, 8428, 8479, 8482, 8708, 8780, 8828, 8842, 8870, 8882, 9134, 9168, 9273, 9277, 9515, 9551, 9581, 9802, 9806, 9848, 9884, 9999}

3.  c222227222772202078  (1723)  {695, 825, 905, 0655, 0857, 0864, 1385, 1857, 2695, 3078, 3285, 3857, 4189, 4857, 5778, 5787, 6054, 6504, 6685, 6856, 6857, 6865, 7864, 8265, 8459, 8572, 9005, 9050, 9857}

4.  c332  (14072)  {323, 332, 3238, 3283, 3328, 5133, 7132}

5.  c3933  (10538)  {333, 3339, 3393, 3933}

6.  c5464644657500000011711019071641751  (49)  {243, 2439, 6782}

Roughly, in order of precursor frequency. Strictly speaking, the bold numbers are not precursors (to cycles) but, rather, cycle terms themselves. I include them for the sake of completeness. Are there any more cycles?

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