[seqfan] Re: Subtracting digits, hitting zero

[Hans Havermann]

Eric Angelini:

> 35 hits 0 when successively subtracting its own "digit-trail"

> 1) What would be the sequence S of such "hitting-zero" integers?
>   I guess S starts like this:
>   S = 0,1,2,3,4,5,6,7,8,9,10,19,20,22,...

To expand the nomenclature slightly, we can talk about the "miss"  
number of a non-negative integer as the negative of the first number  
less than one arrived at by applying the successively-subtracting-its- 
own-digit-trail procedure. As such, miss numbers range from 0 to 8.  
I've put a b-file of miss numbers here:

http://chesswanks.com/num/miss.txt

More specifically, you are asking for numbers with miss of 0:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 22, 28, 29, 31, 32, 34, 35,  
43, 44, 46, 52, 53, 59, 62, 64, 65, 68, 74, 77, 79, 82, 83, 88, 100,  
105, 119, 121, 147, 150, 154, 159, 161, 162, 164, 178, 183, 186, 187,  
192, 193, 196, 198, 208, 209, 214, 219, 220, 231, 233, 238, 243, 244,  
257, 269, 278, 280, 290, 294, 298, 305, 313, 314, 315, 318, 321, 326,  
331, 336, 341, 342, 350, 354, 358, 360, 366, 367, 368, 372, 373, 374,  
377, 379, 386, 394, 396, 402, 403, 416, 424, 426, 427, 429, 430, 431,  
443, 445, 448, 452, 456, 457, 465, 466, 468, 469, 471, 472, 474, 475,  
484, 486, 487, 494, 497, 498, 503, 509, 510, 512, 518, 522, 523, 526,  
528, 529, 536, 543, 546, 550, 555, 558, 568, 571, 582, 598, 600, 602,  
604, 608, 609, 611, 615, 616, 620, 627, 629, 631, 636, 642, 650, 651,  
656, 657, 658, 659, 666, 668, 672, 674, 680, 683, 684, 688, 689, 691,  
696, 706, 707, 712, 715, 718, 740, 747, 754, 756, 757, 762, 763, 775,  
785, 788, 794, 795, 803, 804, 809, 816, 818, 819, 824, 831, 840, 844,  
854, 858, 863, 872, 874, 877, 889, 892, 899, 905, 906, 911, 915, 918,  
921, 922, 933, 934, 935, 942, 944, 948, 954, 956, 957, 960, 964, 966,  
971, 976, 979, 980, 994, 1006, 1008, 1009, 1010, 1012, 1018, 1020,  
1028, 1033, 1043, 1060, 1080, 1086, 1089, 1098, 1107, 1121, 1123,  
1132, 1133, 1143, 1145, 1153, 1160, 1165, 1169, 1170, 1171, 1174,  
1175, 1181, 1186, 1191, 1195, 1196, 1197, 1199, 1200, 1204, 1206,  
1215, 1218, 1235, 1237, 1241, 1245, 1252, 1257, 1265, 1268, 1271,  
1282, 1285, 1289, 1297, 1306, 1310, 1332, 1341, 1343, 1354, 1355,  
1356, 1359, 1365, 1371, 1387, 1392, 1400, 1401, 1404, 1409, 1413,  
1421, 1422, 1424, 1431, 1433, 1436, 1442, 1445, 1447, 1454, 1456,  
1459, 1462, 1464, 1466, 1467, 1468, 1476, 1478, 1481, 1482, 1484,  
1485, 1488, 1490, 1495, 1496, 1500, 1505, 1511, 1514, 1517, 1523,  
1525, 1531, 1535, 1541, 1543, 1547, 1549, 1551, 1552, 1554, 1564,  
1566, 1571, 1574, 1576, 1578, 1580, 1582, 1584, 1585, 1586, 1588,  
1589, 1593, 1597, 1598, 1600, 1601, 1602, 1603, 1606, 1608, 1619,  
1624, 1626, 1630, 1632, 1636, 1637, 1642, 1646, 1648, 1649, 1652,  
1661, 1663, 1664, 1666, 1668, 1669, 1670, 1672, 1673, 1675, 1679,  
1680, 1685, 1686, 1687, 1690, 1695, 1697, 1703, 1704, 1705, 1709,  
1710, 1713, 1716, 1719, 1722, 1725, 1726, 1727, 1734, 1739, 1740,  
1741, 1743, 1744, 1748, 1750, 1752, 1754, 1761, 1764, 1768, 1774,  
1778, 1792, 1793, 1796, 1798, 1799, 1800, 1814, 1817, 1819, 1823,  
1824, 1829, 1830, 1832, 1840, 1845, 1846, 1847, 1848, 1853, 1856,  
1857, 1858, 1861, 1862, 1868, 1869, 1871, 1872, 1876, 1882, 1883,  
1890, 1894, 1895, 1901, 1905, 1914, 1918, 1919, 1922, 1926, 1927,  
1930, 1931, 1932, 1933, 1937, 1944, 1948, 1955, 1957, 1961, 1966,  
1968, 1974, 1977, 1978, 1980, 1981, 1982, 1983, 1985, 1987, 1988,  
1991, 1993, 1997, 2000, 2002, 2003, 2007, 2012, 2015, 2016, 2019,  
2020, 2024, 2026, 2028, 2031, 2034, 2039, 2048, 2061, 2067, 2073,  
2075, 2083, 2084, 2088, 2093, 2095, 2098, 2111, 2115, 2118, 2121,  
2126, 2127, 2133, 2152, 2155, 2157, 2158, 2159, 2165, 2171, 2176,  
2186, 2190, 2193, 2195, 2196, 2201, 2204, 2212, 2217, 2233, 2236,  
2242, 2243, 2257, 2258, 2269, 2270, 2274, 2275, 2282, 2288, 2296,  
2297, 2302, 2308, 2315, 2322, 2325, 2326, 2332, 2335, 2340, 2359,  
2372, 2373, 2387, 2395, 2398, 2404, 2408, 2415, 2417, 2445, 2449,  
2452, 2457, 2463, 2464, 2466, 2483, 2486, 2488, 2494, 2495, 2496,  
2507, 2510, 2511, 2515, 2520, 2521, 2524, 2525, 2526, 2527, 2528,  
2529, 2540, 2541, 2544, 2558, 2563, 2573, 2577, 2583, 2584, 2586,  
2598, 2617, 2621, 2625, 2627, 2629, 2634, 2639, 2653, 2672, 2673,  
2675, 2683, 2688, 2691, 2696, 2698, 2699, 2702, 2704, 2711, 2716,  
2723, 2724, 2727, 2733, 2734, 2739, 2741, 2742, 2745, 2753, 2754,  
2756, 2766, 2767, 2773, 2774, 2777, 2781, 2785, 2789, 2794, 2797,  
2804, 2807, 2813, 2818, 2823, 2833, 2847, 2850, 2856, 2858, 2859,  
2865, 2868, 2869, 2870, 2872, 2875, 2893, 2901, 2903, 2905, 2908,  
2915, 2920, 2924, 2927, 2928, 2932, 2940, 2941, 2942, 2943, 2947,  
2953, 2958, 2966, 2967, 2971, 2972, 2975, 2976, 2979, 2983, 2990,  
2991, 2993, 3008, 3014, 3017, 3020, 3022, 3026, 3031, 3035, 3036,  
3040, 3041, 3046, 3054, 3057, 3058, 3060, 3069, 3070, 3082, 3085,  
3087, 3089, 3090, 3091, 3094, 3097, 3102, 3103, 3112, 3115, 3116,  
3117, 3122, 3123, 3127, 3128, 3129, 3132, 3137, 3140, 3155, 3161,  
3162, 3165, 3167, 3174, 3181, 3182, 3188, 3190, 3191, 3192, 3201,  
3203, 3210, 3220, 3227, 3241, 3242, 3244, 3254, 3269, 3273, 3276,  
3277, 3281, 3282, 3283, 3290, 3302, 3303, 3307, 3309, 3322, 3328,  
3329, 3333, 3337, 3346, 3348, 3349, 3356, 3357, 3358, 3361, 3362,  
3365, 3366, 3367, 3368, 3370, 3373, 3375, 3376, 3383, 3385, 3387,  
3393, 3396, 3398, 3399, 3401, 3408, 3411, 3422, 3428, 3430, 3431,  
3437, 3442, 3448, 3452, 3455, 3456, 3457, 3462, 3463, 3465, 3467,  
3468, 3473, 3478, 3480, 3490, 3491, 3506, 3510, 3513, 3514, 3516,  
3529, 3539, 3544, 3546, 3551, 3554, 3558, 3562, 3565, 3582, 3583,  
3585, 3591, 3594, 3595, 3604, 3605, 3608, 3621, 3627, 3629, 3630,  
3631, 3633, 3639, 3641, 3646, 3657, 3658, 3660, 3668, 3678, 3680,  
3686, 3688, 3694, 3700, 3701, 3703, 3705, 3708, 3711, 3713, 3714,  
3717, 3719, 3727, 3728, 3738, 3746, 3747, 3751, 3754, 3759, 3762,  
3771, 3779, 3783, 3795, 3796, 3797, 3798, 3799, 3801, 3802, 3805,  
3816, 3818, 3819, 3820, 3822, 3830, 3836, 3842, 3847, 3848, 3852,  
3858, 3861, 3866, 3868, 3879, 3881, 3884, 3886, 3890, 3893, 3895,  
3902, 3909, 3916, 3917, 3930, 3931, 3940, 3942, 3946, 3957, 3961,  
3971, 3972, 3973, 3981, 3984, 3986, 3993, 3995, 3996, 3998, 4001,  
4003, 4008, 4014, 4026, 4027, 4029, 4039, 4045, 4057, 4085, 4092,  
4107, 4112, 4117, 4126, 4134, 4135, 4148, 4151, 4154, 4157, 4162,  
4163, 4168, 4169, 4174, 4176, 4179, 4183, 4188, 4189, 4193, 4197,  
4201, 4205, 4206, 4210, 4211, 4221, 4226, 4230, 4234, 4243, 4248,  
4250, 4251, 4258, 4266, 4275, 4277, 4284, 4287, 4288, 4306, 4313,  
4320, 4325, 4326, 4327, 4333, 4343, 4346, 4349, 4351, 4354, 4357,  
4362, 4367, 4371, 4375, 4377, 4385, 4387, 4388, 4401, 4410, 4414,  
4420, 4421, 4424, 4426, 4431, 4438, 4439, 4441, 4443, 4454, 4455,  
4461, 4463, 4464, 4467, 4469, 4477, 4480, 4484, 4487, 4490, 4491,  
4496, 4497, 4498, 4500, 4505, 4508, 4509, 4510, 4515, 4516, 4517,  
4519, 4524, 4526, 4530, 4533, 4541, 4548, 4550, 4558, 4560, 4573,  
4580, 4582, 4585, 4592, 4595, 4597, 4618, 4619, 4622, 4623, 4628,  
4633, 4635, 4646, 4647, 4649, 4651, 4652, 4655, 4657, 4662, 4663,  
4667, 4672, 4679, 4683, 4695, 4700, 4701, 4703, 4704, 4705, 4711,  
4712, 4714, 4715, 4716, 4718, 4720, 4722, 4725, 4726, 4731, 4732,  
4733, 4734, 4738, 4749, 4751, 4757, 4759, 4763, 4765, 4767, 4768,  
4777, 4779, 4789, 4797, 4799, 4801, 4803, 4809, 4818, 4821, 4824,  
4830, 4832, 4833, 4834, 4835, 4840, 4844, 4847, 4854, 4856, 4860,  
4862, 4863, 4870, 4877, 4886, 4887, 4889, 4894, 4899, 4905, 4910,  
4920, 4928, 4932, 4937, 4941, 4944, 4949, 4951, 4954, 4955, 4956,  
4958, 4967, 4978, 4991, 5001, 5006, 5007, 5013, 5015, 5025, 5028,  
5032, 5036, 5044, 5054, 5060, 5066, 5067, 5073, 5074, 5080, 5081,  
5084, 5085, 5088, 5093, 5094, 5095, 5104, 5110, 5113, 5121, 5126,  
5128, 5130, 5134, 5139, 5149, 5160, 5163, 5167, 5171, 5184, 5191,  
5194, 5201, 5202, 5217, 5224, 5225, 5228, 5233, 5243, 5246, 5248,  
5252, 5259, 5260, 5261, 5265, 5267, 5268, 5277, 5281, 5284, 5286,  
5291, 5292, 5297, 5304, 5310, 5312, 5317, 5323, 5328, 5330, 5332,  
5344, 5351, 5365, 5369, 5370, 5374, 5389, 5398, 5400, 5407, 5414,  
5415, 5419, 5421, 5427, 5439, 5440, 5441, 5443, 5450, 5460, 5461,  
5465, 5471, 5472, 5473, 5474, 5477, 5480, 5483, 5484, 5497, 5498,  
5502, 5503, 5504, 5505, 5508, 5509, 5511, 5515, 5518, 5523, 5525,  
5527, 5531, 5537, 5539, 5541, 5543, 5547, 5548, 5549, 5555, 5559,  
5560, 5562, 5568, 5570, 5571, 5574, 5582, 5584, 5587, 5598, 5601,  
5605, 5606, 5607, 5608, 5612, 5616, 5619, 5622, 5626, 5629, 5630,  
5633, 5638, 5639, 5641, 5646, 5647, 5648, 5649, 5657, 5659, 5660,  
5664, 5665, 5666, 5667, 5670, 5677, 5679, 5681, 5683, 5689, 5690,  
5692, 5695, 5699, 5701, 5705, 5708, 5709, 5710, 5711, 5715, 5724,  
5725, 5726, 5728, 5734, 5737, 5739, 5742, 5745, 5746, 5749, 5755,  
5760, 5772, 5776, 5777, 5779, 5782, 5789, 5795, 5799, 5803, 5805,  
5818, 5820, 5821, 5824, 5830, 5831, 5832, 5836, 5838, 5840, 5845,  
5847, 5849, 5850, 5855, 5856, 5857, 5858, 5866, 5880, 5882, 5884,  
5890, 5892, 5893, 5900, 5904, 5911, 5913, 5914, 5918, 5920, 5924,  
5925, 5933, 5934, 5939, 5940, 5941, 5946, 5953, 5955, 5959, 5961,  
5966, 5967, 5968, 5969, 5987, 5993, 5996, 6013, 6015, 6019, 6020,  
6024, 6027, 6031, 6034, 6036, 6047, 6056, 6063, 6066, 6073, 6074,  
6077, 6103, 6106, 6118, 6122, 6123, 6134, 6139, 6146, 6155, 6158,  
6161, 6164, 6173, 6178, 6179, 6181, 6184, 6193, 6197, 6205, 6212,  
6213, 6214, 6223, 6230, 6235, 6239, 6248, 6253, 6264, 6271, 6275,  
6280, 6282, 6288, 6289, 6290, 6303, 6304, 6312, 6322, 6324, 6330,  
6332, 6343, 6346, 6352, 6353, 6355, 6357, 6358, 6368, 6369, 6377,  
6384, 6388, 6404, 6415, 6423, 6426, 6431, 6437, 6440, 6447, 6455,  
6459, 6462, 6463, 6466, 6475, 6479, 6487, 6488, 6490, 6495, 6498,  
6499, 6510, 6513, 6516, 6520, 6522, 6532, 6545, 6549, 6560, 6567,  
6577, 6591, 6592, 6596, 6604, 6610, 6624, 6632, 6634, 6635, 6644,  
6649, 6658, 6671, 6674, 6676, 6677, 6678, 6679, 6681, 6687, 6689,  
6694, 6707, 6716, 6720, 6724, 6725, 6726, 6733, 6738, 6741, 6748,  
6752, 6756, 6781, 6782, 6788, 6790, 6795, 6806, 6809, 6810, 6812,  
6813, 6817, 6818, 6826, 6831, 6832, 6834, 6837, 6846, 6851, 6853,  
6857, 6860, 6864, 6865, 6874, 6879, 6885, 6894, 6897, 6898, 6903,  
6905, 6906, 6911, 6914, 6918, 6924, 6927, 6928, 6929, 6939, 6959,  
6962, 6969, 6976, 6982, 6984, 6989, 6995, 6996, 7027, 7033, 7043,  
7044, 7055, 7064, 7065, 7069, 7070, 7073, 7077, 7079, 7082, 7086,  
7087, 7093, 7097, 7098, 7106, 7109, 7115, 7116, 7117, 7119, 7120,  
7121, 7132, 7133, 7135, 7137, 7151, 7153, 7154, 7159, 7163, 7165,  
7167, 7169, 7171, 7179, 7181, 7185, 7189, 7202, 7205, 7206, 7208,  
7211, 7214, 7223, 7234, 7237, 7243, 7247, 7248, 7249, 7258, 7259,  
7262, 7268, 7270, 7275, 7276, 7282, 7285, 7289, 7295, 7298, 7302,  
7316, 7319, 7326, 7328, 7331, 7333, 7338, 7342, 7348, 7353, 7358,  
7373, 7375, 7389, 7394, 7398, 7400, 7403, 7410, 7425, 7428, 7434,  
7442, 7456, 7461, 7465, 7479, 7489, 7491, 7498, 7501, 7506, 7513,  
7537, 7546, 7553, 7562, 7574, 7581, 7585, 7586, 7589, 7598, 7600,  
7601, 7606, 7607, 7612, 7620, 7628, 7629, 7634, 7639, 7642, 7648,  
7652, 7662, 7664, 7670, 7675, 7676, 7680, 7688, 7690, 7691, 7699,  
7703, 7708, 7714, 7720, 7721, 7734, 7735, 7738, 7750, 7754, 7764,  
7765, 7766, 7785, 7794, 7796, 7797, 7803, 7806, 7817, 7820, 7822,  
7823, 7826, 7834, 7850, 7857, 7861, 7871, 7876, 7879, 7887, 7889,  
7893, 7894, 7899, 7906, 7916, 7933, 7939, 7940, 7941, 7953, 7954,  
7969, 7973, 7975, 7979, 7984, 7988, 7996, 7997, 8003, 8008, 8022,  
8027, 8034, 8038, 8046, 8054, 8056, 8059, 8062, 8063, 8065, 8071,  
8080, 8084, 8094, 8097, 8099, 8101, 8110, 8112, 8115, 8118, 8126,  
8134, 8137, 8161, 8178, 8179, 8181, 8182, 8189, 8192, 8194, 8195,  
8202, 8206, 8208, 8215, 8220, 8229, 8231, 8233, 8234, 8244, 8245,  
8246, 8247, 8248, 8250, 8258, 8260, 8261, 8265, 8270, 8271, 8278,  
8282, 8285, 8289, 8301, 8303, 8308, 8320, 8321, 8332, 8336, 8342,  
8343, 8352, 8357, 8359, 8364, 8372, 8373, 8379, 8381, 8388, 8393,  
8395, 8409, 8412, 8415, 8417, 8419, 8424, 8430, 8439, 8444, 8445,  
8448, 8458, 8461, 8481, 8486, 8500, 8502, 8506, 8518, 8524, 8528,  
8531, 8545, 8546, 8548, 8555, 8572, 8580, 8585, 8586, 8592, 8595,  
8598, 8601, 8611, 8612, 8618, 8623, 8625, 8627, 8629, 8637, 8639,  
8643, 8655, 8658, 8660, 8663, 8665, 8674, 8675, 8676, 8684, 8692,  
8700, 8701, 8704, 8716, 8722, 8727, 8728, 8732, 8734, 8743, 8747,  
8751, 8760, 8769, 8772, 8776, 8787, 8789, 8793, 8800, 8804, 8805,  
8812, 8818, 8819, 8821, 8822, 8832, 8833, 8834, 8838, 8840, 8844,  
8847, 8852, 8864, 8876, 8879, 8881, 8882, 8884, 8887, 8889, 8894,  
8902, 8903, 8915, 8933, 8934, 8937, 8941, 8945, 8947, 8950, 8952,  
8959, 8960, 8966, 8969, 8977, 8984, 8996, 8997, 9002, 9003, 9013,  
9015, 9016, 9021, 9024, 9028, 9031, 9041, 9043, 9045, 9055, 9057,  
9086, 9092, 9095, 9097, 9099, 9102, 9103, 9106, 9110, 9111, 9122,  
9123, 9129, 9135, 9140, 9143, 9147, 9150, 9163, 9166, 9169, 9170,  
9175, 9180, 9184, 9187, 9189, 9210, 9211, 9214, 9217, 9218, 9223,  
9234, 9239, 9249, 9254, 9255, 9274, 9283, 9287, 9289, 9290, 9299,  
9309, 9313, 9314, 9322, 9335, 9340, 9341, 9352, 9361, 9362, 9373,  
9388, 9402, 9409, 9430, 9447, 9464, 9470, 9473, 9493, 9511, 9513,  
9520, 9521, 9525, 9531, 9541, 9544, 9547, 9567, 9580, 9588, 9591,  
9597, 9604, 9615, 9616, 9618, 9631, 9637, 9641, 9656, 9659, 9662,  
9663, 9677, 9680, 9684, 9686, 9695, 9709, 9710, 9713, 9722, 9724,  
9726, 9729, 9731, 9735, 9738, 9742, 9743, 9745, 9748, 9754, 9757,  
9771, 9776, 9777, 9779, 9786, 9798, 9804, 9808, 9813, 9818, 9829,  
9835, 9850, 9860, 9862, 9867, 9870, 9876, 9877, 9883, 9886, 9907,  
9913, 9916, 9917, 9927, 9929, 9935, 9940, 9945, 9950, 9951, 9954,  
9957, 9959, 9961, 9963, 9964, 9965, 9975, 9976, 9992}