Matematické Fórum

lukaszh · 16. 02. 2008 11:24

Neviem prist na sposob ako zistit pocet cifier napriklad cisla: 123456^1005.
Ja som taketo priklady skusal riesit postupnostami.
Napriklad 123456^1 ma 6 cifier; 123456^2 ma 11 cifier. Doplnil som to do aritmetickej postupnosti s diferenciou 5. Potom a_1005 = 6 + (1005-1)*5.
Vyslo mi ze to cislo ma 5026 cifier. No neviem to overit.

Saturday

Dal jsem to do Mathematici a výsledek je 5117.

EDIT: tady je tabulka s počty cifer čísla: 123456^i:

{6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 57, 62, 67, 72, 77, 82, 87, \
92, 97, 102, 107, 113, 118, 123, 128, 133, 138, 143, 148, 153, 158, \
163, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 225, 230, \
235, 240, 245, 250, 255, 260, 265, 270, 275, 281, 286, 291, 296, 301, \
306, 311, 316, 321, 326, 331, 337, 342, 347, 352, 357, 362, 367, 372, \
377, 382, 387, 393, 398, 403, 408, 413, 418, 423, 428, 433, 438, 443, \
449, 454, 459, 464, 469, 474, 479, 484, 489, 494, 499, 505, 510, 515, \
520, 525, 530, 535, 540, 545, 550, 555, 561, 566, 571, 576, 581, 586, \
591, 596, 601, 606, 611, 617, 622, 627, 632, 637, 642, 647, 652, 657, \
662, 667, 673, 678, 683, 688, 693, 698, 703, 708, 713, 718, 723, 729, \
734, 739, 744, 749, 754, 759, 764, 769, 774, 780, 785, 790, 795, 800, \
805, 810, 815, 820, 825, 830, 836, 841, 846, 851, 856, 861, 866, 871, \
876, 881, 886, 892, 897, 902, 907, 912, 917, 922, 927, 932, 937, 942, \
948, 953, 958, 963, 968, 973, 978, 983, 988, 993, 998, 1004, 1009, \
1014, 1019, 1024, 1029, 1034, 1039, 1044, 1049, 1054, 1060, 1065, \
1070, 1075, 1080, 1085, 1090, 1095, 1100, 1105, 1110, 1116, 1121, \
1126, 1131, 1136, 1141, 1146, 1151, 1156, 1161, 1166, 1172, 1177, \
1182, 1187, 1192, 1197, 1202, 1207, 1212, 1217, 1222, 1228, 1233, \
1238, 1243, 1248, 1253, 1258, 1263, 1268, 1273, 1278, 1284, 1289, \
1294, 1299, 1304, 1309, 1314, 1319, 1324, 1329, 1334, 1340, 1345, \
1350, 1355, 1360, 1365, 1370, 1375, 1380, 1385, 1390, 1396, 1401, \
1406, 1411, 1416, 1421, 1426, 1431, 1436, 1441, 1446, 1452, 1457, \
1462, 1467, 1472, 1477, 1482, 1487, 1492, 1497, 1502, 1508, 1513, \
1518, 1523, 1528, 1533, 1538, 1543, 1548, 1553, 1559, 1564, 1569, \
1574, 1579, 1584, 1589, 1594, 1599, 1604, 1609, 1615, 1620, 1625, \
1630, 1635, 1640, 1645, 1650, 1655, 1660, 1665, 1671, 1676, 1681, \
1686, 1691, 1696, 1701, 1706, 1711, 1716, 1721, 1727, 1732, 1737, \
1742, 1747, 1752, 1757, 1762, 1767, 1772, 1777, 1783, 1788, 1793, \
1798, 1803, 1808, 1813, 1818, 1823, 1828, 1833, 1839, 1844, 1849, \
1854, 1859, 1864, 1869, 1874, 1879, 1884, 1889, 1895, 1900, 1905, \
1910, 1915, 1920, 1925, 1930, 1935, 1940, 1945, 1951, 1956, 1961, \
1966, 1971, 1976, 1981, 1986, 1991, 1996, 2001, 2007, 2012, 2017, \
2022, 2027, 2032, 2037, 2042, 2047, 2052, 2057, 2063, 2068, 2073, \
2078, 2083, 2088, 2093, 2098, 2103, 2108, 2113, 2119, 2124, 2129, \
2134, 2139, 2144, 2149, 2154, 2159, 2164, 2169, 2175, 2180, 2185, \
2190, 2195, 2200, 2205, 2210, 2215, 2220, 2225, 2231, 2236, 2241, \
2246, 2251, 2256, 2261, 2266, 2271, 2276, 2281, 2287, 2292, 2297, \
2302, 2307, 2312, 2317, 2322, 2327, 2332, 2338, 2343, 2348, 2353, \
2358, 2363, 2368, 2373, 2378, 2383, 2388, 2394, 2399, 2404, 2409, \
2414, 2419, 2424, 2429, 2434, 2439, 2444, 2450, 2455, 2460, 2465, \
2470, 2475, 2480, 2485, 2490, 2495, 2500, 2506, 2511, 2516, 2521, \
2526, 2531, 2536, 2541, 2546, 2551, 2556, 2562, 2567, 2572, 2577, \
2582, 2587, 2592, 2597, 2602, 2607, 2612, 2618, 2623, 2628, 2633, \
2638, 2643, 2648, 2653, 2658, 2663, 2668, 2674, 2679, 2684, 2689, \
2694, 2699, 2704, 2709, 2714, 2719, 2724, 2730, 2735, 2740, 2745, \
2750, 2755, 2760, 2765, 2770, 2775, 2780, 2786, 2791, 2796, 2801, \
2806, 2811, 2816, 2821, 2826, 2831, 2836, 2842, 2847, 2852, 2857, \
2862, 2867, 2872, 2877, 2882, 2887, 2892, 2898, 2903, 2908, 2913, \
2918, 2923, 2928, 2933, 2938, 2943, 2948, 2954, 2959, 2964, 2969, \
2974, 2979, 2984, 2989, 2994, 2999, 3004, 3010, 3015, 3020, 3025, \
3030, 3035, 3040, 3045, 3050, 3055, 3060, 3066, 3071, 3076, 3081, \
3086, 3091, 3096, 3101, 3106, 3111, 3117, 3122, 3127, 3132, 3137, \
3142, 3147, 3152, 3157, 3162, 3167, 3173, 3178, 3183, 3188, 3193, \
3198, 3203, 3208, 3213, 3218, 3223, 3229, 3234, 3239, 3244, 3249, \
3254, 3259, 3264, 3269, 3274, 3279, 3285, 3290, 3295, 3300, 3305, \
3310, 3315, 3320, 3325, 3330, 3335, 3341, 3346, 3351, 3356, 3361, \
3366, 3371, 3376, 3381, 3386, 3391, 3397, 3402, 3407, 3412, 3417, \
3422, 3427, 3432, 3437, 3442, 3447, 3453, 3458, 3463, 3468, 3473, \
3478, 3483, 3488, 3493, 3498, 3503, 3509, 3514, 3519, 3524, 3529, \
3534, 3539, 3544, 3549, 3554, 3559, 3565, 3570, 3575, 3580, 3585, \
3590, 3595, 3600, 3605, 3610, 3615, 3621, 3626, 3631, 3636, 3641, \
3646, 3651, 3656, 3661, 3666, 3671, 3677, 3682, 3687, 3692, 3697, \
3702, 3707, 3712, 3717, 3722, 3727, 3733, 3738, 3743, 3748, 3753, \
3758, 3763, 3768, 3773, 3778, 3783, 3789, 3794, 3799, 3804, 3809, \
3814, 3819, 3824, 3829, 3834, 3840, 3845, 3850, 3855, 3860, 3865, \
3870, 3875, 3880, 3885, 3890, 3896, 3901, 3906, 3911, 3916, 3921, \
3926, 3931, 3936, 3941, 3946, 3952, 3957, 3962, 3967, 3972, 3977, \
3982, 3987, 3992, 3997, 4002, 4008, 4013, 4018, 4023, 4028, 4033, \
4038, 4043, 4048, 4053, 4058, 4064, 4069, 4074, 4079, 4084, 4089, \
4094, 4099, 4104, 4109, 4114, 4120, 4125, 4130, 4135, 4140, 4145, \
4150, 4155, 4160, 4165, 4170, 4176, 4181, 4186, 4191, 4196, 4201, \
4206, 4211, 4216, 4221, 4226, 4232, 4237, 4242, 4247, 4252, 4257, \
4262, 4267, 4272, 4277, 4282, 4288, 4293, 4298, 4303, 4308, 4313, \
4318, 4323, 4328, 4333, 4338, 4344, 4349, 4354, 4359, 4364, 4369, \
4374, 4379, 4384, 4389, 4394, 4400, 4405, 4410, 4415, 4420, 4425, \
4430, 4435, 4440, 4445, 4450, 4456, 4461, 4466, 4471, 4476, 4481, \
4486, 4491, 4496, 4501, 4506, 4512, 4517, 4522, 4527, 4532, 4537, \
4542, 4547, 4552, 4557, 4562, 4568, 4573, 4578, 4583, 4588, 4593, \
4598, 4603, 4608, 4613, 4619, 4624, 4629, 4634, 4639, 4644, 4649, \
4654, 4659, 4664, 4669, 4675, 4680, 4685, 4690, 4695, 4700, 4705, \
4710, 4715, 4720, 4725, 4731, 4736, 4741, 4746, 4751, 4756, 4761, \
4766, 4771, 4776, 4781, 4787, 4792, 4797, 4802, 4807, 4812, 4817, \
4822, 4827, 4832, 4837, 4843, 4848, 4853, 4858, 4863, 4868, 4873, \
4878, 4883, 4888, 4893, 4899, 4904, 4909, 4914, 4919, 4924, 4929, \
4934, 4939, 4944, 4949, 4955, 4960, 4965, 4970, 4975, 4980, 4985, \
4990, 4995, 5000, 5005, 5011, 5016, 5021, 5026, 5031, 5036, 5041, \
5046, 5051, 5056, 5061, 5067, 5072, 5077, 5082, 5087, 5092, 5097, \
5102, 5107, 5112, 5117}

podívej se na rozdíl mezi 10. a 11. členem...

takže: ten tvůj postup + zaokrouhlit_dolu(1005/11) = 5117

nyní ještě, proč to tak funguje :-))

Kondr · 16. 02. 2008 15:00

Možná ještě jednu poznámku: počet cifer přirozeného čísla k je
$\lfloor log_{10}k\rfloor+1$ , v našem případě
$X=\lfloor log_{10}(123456^{1005})\rfloor+1= \lfloor log_{10}(123456)\cdot 1005\rfloor+1=\nl =\lfloor \left(5+log_{10}(1,23456)\right)\cdot 1005\rfloor+1= \lfloor log_{10}(1,23456)\cdot 1005\rfloor+5\cdot1005+1$

A ty divné závorky, to je zaokrouhlení dolů. Desítkový logaritmus nám zde stačí na 4 desetinná místa, což zvládne každá kalkulačka.

Matematické Fórum

#1 16. 02. 2008 11:24

pocet cifier

#2 16. 02. 2008 13:07 — Editoval Saturday (16. 02. 2008 13:26)

Re: pocet cifier

#3 16. 02. 2008 15:00

Re: pocet cifier

Zápatí