Сума дільників числа

Сума делителей числа. Для початок наведемо експериментальний матеріал (який було отримано з допомогою програми Derive (за такою формулою 1.(см.ниже)): перебування делителей числа «a», програма ділила число «a» інші числа не переважали саме число і якщо залишок від розподілу дорівнював 0, то число записувалося як дільник «a».): Нижче наведені все делители чисел від 1 до 1000: [1, [1]] [2, [1, 2]] [3, [1, 3]] [4, [1, 2, 4]] [5, [1, 5]] [6, [1, 2, 3, 6]] [7, [1, 7]] [8, [1, 2, 4, 8]] [9, [1, 3, 9]] [10, [1, 2, 5, 10]] [11, [1, 11]] [12, [1, 2, 3, 4, 6, 12]] [13, [1, 13]] [14, [1, 2, 7, 14]] [15, [1, 3, 5, 15]] [16, [1, 2, 4, 8, 16]] [17, [1, 17]] [18, [1, 2, 3, 6, 9, 18]] [19, [1, 19]] [20, [1, 2, 4, 5, 10, 20]] [21, [1, 3, 7, 21]] [22, [1, 2, 11, 22]] [23, [1, 23]] [24, [1, 2, 3, 4, 6, 8, 12, 24]] [25, [1, 5, 25]] [26, [1, 2, 13, 26]] [27, [1, 3, 9, 27]] [28, [1, 2, 4, 7, 14, 28]] [29, [1, 29]] [30, [1, 2, 3, 5, 6, 10, 15, 30]] [31, [1, 31]] [32, [1, 2, 4, 8, 16, 32]] [33, [1, 3, 11, 33]] [34, [1, 2, 17, 34]] [35, [1, 5, 7, 35]] [36, [1, 2, 3, 4, 6, 9, 12, 18, 36]] [37, [1, 37]] [38, [1, 2, 19, 38]] [39, [1, 3, 13, 39]] [40, [1, 2, 4, 5, 8, 10, 20, 40]] [41, [1, 41]] [42, [1, 2, 3, 6, 7, 14, 21, 42]] [43, [1, 43]] [44, [1, 2, 4, 11, 22, 44]] [45, [1, 3, 5, 9, 15, 45]] [46, [1, 2, 23, 46]] [47, [1, 47]] [48, [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]] [49, [1, 7, 49]] [50, [1, 2, 5, 10, 25, 50]] [51, [1, 3, 17, 51]] [52, [1, 2, 4, 13, 26, 52]] [53, [1, 53]] [54, [1, 2, 3, 6, 9, 18, 27, 54]] [55, [1, 5, 11, 55]] [56, [1, 2, 4, 7, 8, 14, 28, 56]] [57, [1, 3, 19, 57]] [58, [1, 2, 29, 58]] [59, [1, 59]] [60, [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]] [61, [1, 61]] [62, [1, 2, 31, 62]] [63, [1, 3, 7, 9, 21, 63]] [64, [1, 2, 4, 8, 16, 32, 64]] [65, [1, 5, 13, 65]] [66, [1, 2, 3, 6, 11, 22, 33, 66]] [67, [1, 67]] [68, [1, 2, 4, 17, 34, 68]] [69, [1, 3, 23, 69]] [70, [1, 2, 5, 7, 10, 14, 35, 70]] [71, [1, 71]] [72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]] [73, [1, 73]] [74, [1, 2, 37, 74]] [75, [1, 3, 5, 15, 25, 75]] [76, [1, 2, 4, 19, 38, 76]] [77, [1, 7, 11, 77]] [78, [1, 2, 3, 6, 13, 26, 39, 78]] [79, [1, 79]] [80, [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]] [81, [1, 3, 9, 27, 81]] [82, [1, 2, 41, 82]] [83, [1, 83]] [84, [1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84]] [85, [1, 5, 17, 85]] [86, [1, 2, 43, 86]] [87, [1, 3, 29, 87]] [88, [1, 2, 4, 8, 11, 22, 44, 88]] [89, [1, 89]] [90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90]] [91, [1, 7, 13, 91]] [92, [1, 2, 4, 23, 46, 92]] [93, [1, 3, 31, 93]] [94, [1, 2, 47, 94]] [95, [1, 5, 19, 95]] [96, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96]] [97, [1, 97]] [98, [1, 2, 7, 14, 49, 98]] [99, [1, 3, 9, 11, 33, 99]] [100, [1, 2, 4, 5, 10, 20, 25, 50, 100]] [101, [1, 101]] [102, [1, 2, 3, 6, 17, 34, 51, 102]] [103, [1, 103]] [104, [1, 2, 4, 8, 13, 26, 52, 104]] [105, [1, 3, 5, 7, 15, 21, 35, 105]] [106, [1, 2, 53, 106]] [107, [1, 107]] [108, [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108]] [109, [1, 109]] [110, [1, 2, 5, 10, 11, 22, 55, 110]] [111, [1, 3, 37, 111]] [112, [1, 2, 4, 7, 8, 14, 16, 28, 56, 112]] [113, [1, 113]] [114, [1, 2, 3, 6, 19, 38, 57, 114]] [115, [1, 5, 23, 115]] [116, [1, 2, 4, 29, 58, 116]] [117, [1, 3, 9, 13, 39, 117]] [118, [1, 2, 59, 118]] [119, [1, 7, 17, 119]] [120, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120]] [121, [1, 11, 121]] [122, [1, 2, 61, 122]] [123, [1, 3, 41, 123]] [124, [1, 2, 4, 31, 62, 124]] [125, [1, 5, 25, 125]] [126, [1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126]] [127, [1, 127]] [128, [1, 2, 4, 8, 16, 32, 64, 128]] [129, [1, 3, 43, 129]] [130, [1, 2, 5, 10, 13, 26, 65, 130]] [131, [1, 131]] [132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132]] [133, [1, 7, 19, 133]] [134, [1, 2, 67, 134]] [135, [1, 3, 5, 9, 15, 27, 45, 135]] [136, [1, 2, 4, 8, 17, 34, 68, 136]] [137, [1, 137]] [138, [1, 2, 3, 6, 23, 46, 69, 138]] [139, [1, 139]] [140, [1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140]] [141, [1, 3, 47, 141]] [142, [1, 2, 71, 142]] [143, [1, 11, 13, 143]] [144, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144]] [145, [1, 5, 29, 145]] [146, [1, 2, 73, 146]] [147, [1, 3, 7, 21, 49, 147]] [148, [1, 2, 4, 37, 74, 148]] [149, [1, 149]] [150, [1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150]] [151, [1, 151]] [152, [1, 2, 4, 8, 19, 38, 76, 152]] [153, [1, 3, 9, 17, 51, 153]] [154, [1, 2, 7, 11, 14, 22, 77, 154]] [155, [1, 5, 31, 155]] [156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156]] [157, [1, 157]] [158, [1, 2, 79, 158]] [159, [1, 3, 53, 159]] [160, [1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160]] [161, [1, 7, 23, 161]] [162, [1, 2, 3, 6, 9, 18, 27, 54, 81, 162]] [163, [1, 163]] [164, [1, 2, 4, 41, 82, 164]] [165, [1, 3, 5, 11, 15, 33, 55, 165]] [166, [1, 2, 83, 166]] [167, [1, 167]] [168, [1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168]] [169, [1, 13, 169]] [170, [1, 2, 5, 10, 17, 34, 85, 170]] [171, [1, 3, 9, 19, 57, 171]] [172, [1, 2, 4, 43, 86, 172]] [173, [1, 173]] [174, [1, 2, 3, 6, 29, 58, 87, 174]] [175, [1, 5, 7, 25, 35, 175]] [176, [1, 2, 4, 8, 11, 16, 22, 44, 88, 176]] [177, [1, 3, 59, 177]] [178, [1, 2, 89, 178]] [179, [1, 179]] [180, [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180]] [181, [1, 181]] [182, [1, 2, 7, 13, 14, 26, 91, 182]] [183, [1, 3, 61, 183]] [184, [1, 2, 4, 8, 23, 46, 92, 184]] [185, [1, 5, 37, 185]] [186, [1, 2, 3, 6, 31, 62, 93, 186]] [187, [1, 11, 17, 187]] [188, [1, 2, 4, 47, 94, 188]] [189, [1, 3, 7, 9, 21, 27, 63, 189]] [190, [1, 2, 5, 10, 19, 38, 95, 190]] [191, [1, 191]] [192, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192]] [193, [1, 193]] [194, [1, 2, 97, 194]] [195, [1, 3, 5, 13, 15, 39, 65, 195]] [196, [1, 2, 4, 7, 14, 28, 49, 98, 196]] [197, [1, 197]] [198, [1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198]] [199, [1, 199]] [200, [1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200]] [201, [1, 3, 67, 201]] [202, [1, 2, 101, 202]] [203, [1, 7, 29, 203]] [204, [1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204]] [205, [1, 5, 41, 205]] [206, [1, 2, 103, 206]] [207, [1, 3, 9, 23, 69, 207]] [208, [1, 2, 4, 8, 13, 16, 26, 52, 104, 208]] [209, [1, 11, 19, 209]] [210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210]] [211, [1, 211]] [212, [1, 2, 4, 53, 106, 212]] [213, [1, 3, 71, 213]] [214, [1, 2, 107, 214]] [215, [1, 5, 43, 215]] [216, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216]] [217, [1, 7, 31, 217]] [218, [1, 2, 109, 218]] [219, [1, 3, 73, 219]] [220, [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220]] [221, [1, 13, 17, 221]] [222, [1, 2, 3, 6, 37, 74, 111, 222]] [223, [1, 223]] [224, [1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112, 224]] [225, [1, 3, 5, 9, 15, 25, 45, 75, 225]] [226, [1, 2, 113, 226]] [227, [1, 227]] [228, [1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228]] [229, [1, 229]] [230, [1, 2, 5, 10, 23, 46, 115, 230]] [231, [1, 3, 7, 11, 21, 33, 77, 231]] [232, [1, 2, 4, 8, 29, 58, 116, 232]] [233, [1, 233]] [234, [1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234]] [235, [1, 5, 47, 235]] [236, [1, 2, 4, 59, 118, 236]] [237, [1, 3, 79, 237]] [238, [1, 2, 7, 14, 17, 34, 119, 238]] [239, [1, 239]] [240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240]] [241, [1, 241]] [242, [1, 2, 11, 22, 121, 242]] [243, [1, 3, 9, 27, 81, 243]] [244, [1, 2, 4, 61, 122, 244]] [245, [1, 5, 7, 35, 49, 245]] [246, [1, 2, 3, 6, 41, 82, 123, 246]] [247, [1, 13, 19, 247]] [248, [1, 2, 4, 8, 31, 62, 124, 248]] [249, [1, 3, 83, 249]] [250, [1, 2, 5, 10, 25, 50, 125, 250]] [251, [1, 251]] [252, [1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252]] [253, [1, 11, 23, 253]] [254, [1, 2, 127, 254]] [255, [1, 3, 5, 15, 17, 51, 85, 255]] [256, [1, 2, 4, 8, 16, 32, 64, 128, 256]] [257, [1, 257]] [258, [1, 2, 3, 6, 43, 86, 129, 258]] [259, [1, 7, 37, 259]] [260, [1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260]] [261, [1, 3, 9, 29, 87, 261]] [262, [1, 2, 131, 262]] [263, [1, 263]] [264, [1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264]] [265, [1, 5, 53, 265]] [266, [1, 2, 7, 14, 19, 38, 133, 266]] [267, [1, 3, 89, 267]] [268, [1, 2, 4, 67, 134, 268]] [269, [1, 269]] [270, [1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270]] [271, [1, 271]] [272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272]] [273, [1, 3, 7, 13, 21, 39, 91, 273]] [274, [1, 2, 137, 274]] [275, [1, 5, 11, 25, 55, 275]] [276, [1, 2, 3, 4, 6, 12, 23, 46, 69, 92, 138, 276]] [277, [1, 277]] [278, [1, 2, 139, 278]] [279, [1, 3, 9, 31, 93, 279]] [280, [1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280]] [281, [1, 281]] [282, [1, 2, 3, 6, 47, 94, 141, 282]] [283, [1, 283]] [284, [1, 2, 4, 71, 142, 284]] [285, [1, 3, 5, 15, 19, 57, 95, 285]] [286, [1, 2, 11, 13, 22, 26, 143, 286]] [287, [1, 7, 41, 287]] [288, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288]] [289, [1, 17, 289]] [290, [1, 2, 5, 10, 29, 58, 145, 290]] [291, [1, 3, 97, 291]] [292, [1, 2, 4, 73, 146, 292]] [293, [1, 293]] [294, [1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294]] [295, [1, 5, 59, 295]] [296, [1, 2, 4, 8, 37, 74, 148, 296]] [297, [1, 3, 9, 11, 27, 33, 99, 297]] [298, [1, 2, 149, 298]] [299, [1, 13, 23, 299]] [300, [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300]] [301, [1, 7, 43, 301]] [302, [1, 2, 151, 302]] [303, [1, 3, 101, 303]] [304, [1, 2, 4, 8, 16, 19, 38, 76, 152, 304]] [305, [1, 5, 61, 305]] [306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306]] [307, [1, 307]] [308, [1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308]] [309, [1, 3, 103, 309]] [310, [1, 2, 5, 10, 31, 62, 155, 310]] [311, [1, 311]] [312, [1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312]] [313, [1, 313]] [314, [1, 2, 157, 314]] [315, [1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315]] [316, [1, 2, 4, 79, 158, 316]] [317, [1, 317]] [318, [1, 2, 3, 6, 53, 106, 159, 318]] [319, [1, 11, 29, 319]] [320, [1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 160, 320]] [321, [1, 3, 107, 321]] [322, [1, 2, 7, 14, 23, 46, 161, 322]] [323, [1, 17, 19, 323]] [324, [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324]] [325, [1, 5, 13, 25, 65, 325]] [326, [1, 2, 163, 326]] [327, [1, 3, 109, 327]] [328, [1, 2, 4, 8, 41, 82, 164, 328]] [329, [1, 7, 47, 329]] [330, [1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330]] [331, [1, 331]] [332, [1, 2, 4, 83, 166, 332]] [333, [1, 3, 9, 37, 111, 333]] [334, [1, 2, 167, 334]] [335, [1, 5, 67, 335]] [336, [1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336]] [337, [1, 337]] [338, [1, 2, 13, 26, 169, 338]] [339, [1, 3, 113, 339]] [340, [1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340]] [341, [1, 11, 31, 341]] [342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342]] [343, [1, 7, 49, 343]] [344, [1, 2, 4, 8, 43, 86, 172, 344]] [345, [1, 3, 5, 15, 23, 69, 115, 345]] [346, [1, 2, 173, 346]] [347, [1, 347]] [348, [1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, 348]] [349, [1, 349]] [350, [1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350]] [351, [1, 3, 9, 13, 27, 39, 117, 351]] [352, [1, 2, 4, 8, 11, 16, 22, 32, 44, 88, 176, 352]] [353, [1, 353]] [354, [1, 2, 3, 6, 59, 118, 177, 354]] [355, [1, 5, 71, 355]] [356, [1, 2, 4, 89, 178, 356]] [357, [1, 3, 7, 17, 21, 51, 119, 357]] [358, [1, 2, 179, 358]] [359, [1, 359]] [360, [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360]] [361, [1, 19, 361]] [362, [1, 2, 181, 362]] [363, [1, 3, 11, 33, 121, 363]] [364, [1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182, 364]] [365, [1, 5, 73, 365]] [366, [1, 2, 3, 6, 61, 122, 183, 366]] [367, [1, 367]] [368, [1, 2, 4, 8, 16, 23, 46, 92, 184, 368]] [369, [1, 3, 9, 41, 123, 369]] [370, [1, 2, 5, 10, 37, 74, 185, 370]] [371, [1, 7, 53, 371]] [372, [1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, 372]] [373, [1, 373]] [374, [1, 2, 11, 17, 22, 34, 187, 374]] [375, [1, 3, 5, 15, 25, 75, 125, 375]] [376, [1, 2, 4, 8, 47, 94, 188, 376]] [377, [1, 13, 29, 377]] [378, [1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378]] [379, [1, 379]] [380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380]] [381, [1, 3, 127, 381]] [382, [1, 2, 191, 382]] [383, [1, 383]] [384, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384]] [385, [1, 5, 7, 11, 35, 55, 77, 385]] [386, [1, 2, 193, 386]] [387, [1, 3, 9, 43, 129, 387]] [388, [1, 2, 4, 97, 194, 388]] [389, [1, 389]] [390, [1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, 390]] [391, [1, 17, 23, 391]] [392, [1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, 392]] [393, [1, 3, 131, 393]] [394, [1, 2, 197, 394]] [395, [1, 5, 79, 395]] [396, [1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396]] [397, [1, 397]] [398, [1, 2, 199, 398]] [399, [1, 3, 7, 19, 21, 57, 133, 399]] [400, [1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400]] [401, [1, 401]] [402, [1, 2, 3, 6, 67, 134, 201, 402]] [403, [1, 13, 31, 403]] [404, [1, 2, 4, 101, 202, 404]] [405, [1, 3, 5, 9, 15, 27, 45, 81, 135, 405]] [406, [1, 2, 7, 14, 29, 58, 203, 406]] [407, [1, 11, 37, 407]] [408, [1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408]] [409, [1, 409]] [410, [1, 2, 5, 10, 41, 82, 205, 410]] [411, [1, 3, 137, 411]] [412, [1, 2, 4, 103, 206, 412]] [413, [1, 7, 59, 413]] [414, [1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414]] [415, [1, 5, 83, 415]] [416, [1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416]] [417, [1, 3, 139, 417]] [418, [1, 2, 11, 19, 22, 38, 209, 418]] [419, [1, 419]] [420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420]] [421, [1, 421]] [422, [1, 2, 211, 422]] [423, [1, 3, 9, 47, 141, 423]] [424, [1, 2, 4, 8, 53, 106, 212, 424]] [425, [1, 5, 17, 25, 85, 425]] [426, [1, 2, 3, 6, 71, 142, 213, 426]] [427, [1, 7, 61, 427]] [428, [1, 2, 4, 107, 214, 428]] [429, [1, 3, 11, 13, 33, 39, 143, 429]] [430, [1, 2, 5, 10, 43, 86, 215, 430]] [431, [1, 431]] [432, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432]] [433, [1, 433]] [434, [1, 2, 7, 14, 31, 62, 217, 434]] [435, [1, 3, 5, 15, 29, 87, 145, 435]] [436, [1, 2, 4, 109, 218, 436]] [437, [1, 19, 23, 437]] [438, [1, 2, 3, 6, 73, 146, 219, 438]] [439, [1, 439]] [440, [1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440]] [441, [1, 3, 7, 9, 21, 49, 63, 147, 441]] [442, [1, 2, 13, 17, 26, 34, 221, 442]] [443, [1, 443]] [444, [1, 2, 3, 4, 6, 12, 37, 74, 111, 148, 222, 444]] [445, [1, 5, 89, 445]] [446, [1, 2, 223, 446]] [447, [1, 3, 149, 447]] [448, [1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448]] [449, [1, 449]] [450, [1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450]] [451, [1, 11, 41, 451]] [452, [1, 2, 4, 113, 226, 452]] [453, [1, 3, 151, 453]] [454, [1, 2, 227, 454]] [455, [1, 5, 7, 13, 35, 65, 91, 455]] [456, [1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456]] [457, [1, 457]] [458, [1, 2, 229, 458]] [459, [1, 3, 9, 17, 27, 51, 153, 459]] [460, [1, 2, 4, 5, 10, 20, 23, 46, 92, 115, 230, 460]] [461, [1, 461]] [462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462]] [463, [1, 463]] [464, [1, 2, 4, 8, 16, 29, 58, 116, 232, 464]] [465, [1, 3, 5, 15, 31, 93, 155, 465]] [466, [1, 2, 233, 466]] 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378, 756]] [757, [1, 757]] [758, [1, 2, 379, 758]] [759, [1, 3, 11, 23, 33, 69, 253, 759]] [760, [1, 2, 4, 5, 8, 10, 19, 20, 38, 40, 76, 95, 152, 190, 380, 760]] [761, [1, 761]] [762, [1, 2, 3, 6, 127, 254, 381, 762]] [763, [1, 7, 109, 763]] [764, [1, 2, 4, 191, 382, 764]] [765, [1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765]] [766, [1, 2, 383, 766]] [767, [1, 13, 59, 767]] [768, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768]] [769, [1, 769]] [770, [1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770]] [771, [1, 3, 257, 771]] [772, [1, 2, 4, 193, 386, 772]] [773, [1, 773]] [774, [1, 2, 3, 6, 9, 18, 43, 86, 129, 258, 387, 774]] [775, [1, 5, 25, 31, 155, 775]] [776, [1, 2, 4, 8, 97, 194, 388, 776]] [777, [1, 3, 7, 21, 37, 111, 259, 777]] [778, [1, 2, 389, 778]] [779, [1, 19, 41, 779]] [780, [1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780]] [781, [1, 11, 71, 781]] [782, [1, 2, 17, 23, 34, 46, 391, 782]] [783, [1, 3, 9, 27, 29, 87, 261, 783]] [784, [1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784]] [785, [1, 5, 157, 785]] [786, [1, 2, 3, 6, 131, 262, 393, 786]] [787, [1, 787]] [788, [1, 2, 4, 197, 394, 788]] [789, [1, 3, 263, 789]] [790, [1, 2, 5, 10, 79, 158, 395, 790]] [791, [1, 7, 113, 791]] [792, [1, 2, 3, 4, 6, 8, 9, 11, 12, 18, 22, 24, 33, 36, 44, 66, 72, 88, 99, 132, 198, 264, 396, 792]] [793, [1, 13, 61, 793]] [794, [1, 2, 397, 794]] [795, [1, 3, 5, 15, 53, 159, 265, 795]] [796, [1, 2, 4, 199, 398, 796]] [797, [1, 797]] [798, [1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798]] [799, [1, 17, 47, 799]] [800, [1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800]] [801, [1, 3, 9, 89, 267, 801]] [802, [1, 2, 401, 802]] [803, [1, 11, 73, 803]] [804, [1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804]] [805, [1, 5, 7, 23, 35, 115, 161, 805]] [806, [1, 2, 13, 26, 31, 62, 403, 806]] [807, [1, 3, 269, 807]] [808, [1, 2, 4, 8, 101, 202, 404, 808]] [809, [1, 809]] [810, [1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810]] [811, [1, 811]] [812, [1, 2, 4, 7, 14, 28, 29, 58, 116, 203, 406, 812]] [813, [1, 3, 271, 813]] [814, [1, 2, 11, 22, 37, 74, 407, 814]] [815, [1, 5, 163, 815]] [816, [1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816]] [817, [1, 19, 43, 817]] [818, [1, 2, 409, 818]] [819, [1, 3, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819]] [820, [1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820]] [821, [1, 821]] [822, [1, 2, 3, 6, 137, 274, 411, 822]] [823, [1, 823]] [824, [1, 2, 4, 8, 103, 206, 412, 824]] [825, [1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825]] [826, [1, 2, 7, 14, 59, 118, 413, 826]] [827, [1, 827]] [828, [1, 2, 3, 4, 6, 9, 12, 18, 23, 36, 46, 69, 92, 138, 207, 276, 414, 828]] [829, [1, 829]] [830, [1, 2, 5, 10, 83, 166, 415, 830]] [831, [1, 3, 277, 831]] [832, [1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832]] [833, [1, 7, 17, 49, 119, 833]] [834, [1, 2, 3, 6, 139, 278, 417, 834]] [835, [1, 5, 167, 835]] [836, [1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836]] [837, [1, 3, 9, 27, 31, 93, 279, 837]] [838, [1, 2, 419, 838]] [839, [1, 839]] [840, [1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840]] [841, [1, 29, 841]] [842, [1, 2, 421, 842]] [843, [1, 3, 281, 843]] [844, [1, 2, 4, 211, 422, 844]] [845, [1, 5, 13, 65, 169, 845]] [846, [1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846]] [847, [1, 7, 11, 77, 121, 847]] [848, [1, 2, 4, 8, 16, 53, 106, 212, 424, 848]] [849, [1, 3, 283, 849]] [850, [1, 2, 5, 10, 17, 25, 34, 50, 85, 170, 425, 850]] [851, [1, 23, 37, 851]] [852, [1, 2, 3, 4, 6, 12, 71, 142, 213, 284, 426, 852]] [853, [1, 853]] [854, [1, 2, 7, 14, 61, 122, 427, 854]] [855, [1, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855]] [856, [1, 2, 4, 8, 107, 214, 428, 856]] [857, [1, 857]] [858, [1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858]] [859, [1, 859]] [860, [1, 2, 4, 5, 10, 20, 43, 86, 172, 215, 430, 860]] [861, [1, 3, 7, 21, 41, 123, 287, 861]] [862, [1, 2, 431, 862]] [863, [1, 863]] [864, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864]] [865, [1, 5, 173, 865]] [866, [1, 2, 433, 866]] [867, [1, 3, 17, 51, 289, 867]] [868, [1, 2, 4, 7, 14, 28, 31, 62, 124, 217, 434, 868]] [869, [1, 11, 79, 869]] [870, [1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870]] [871, [1, 13, 67, 871]] [872, [1, 2, 4, 8, 109, 218, 436, 872]] [873, [1, 3, 9, 97, 291, 873]] [874, [1, 2, 19, 23, 38, 46, 437, 874]] [875, [1, 5, 7, 25, 35, 125, 175, 875]] [876, [1, 2, 3, 4, 6, 12, 73, 146, 219, 292, 438, 876]] [877, [1, 877]] [878, [1, 2, 439, 878]] [879, [1, 3, 293, 879]] [880, [1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 40, 44, 55, 80, 88, 110, 176, 220, 440, 880]] [881, [1, 881]] [882, [1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 441, 882]] [883, [1, 883]] [884, [1, 2, 4, 13, 17, 26, 34, 52, 68, 221, 442, 884]] [885, [1, 3, 5, 15, 59, 177, 295, 885]] [886, [1, 2, 443, 886]] [887, [1, 887]] [888, [1, 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, 444, 888]] [889, [1, 7, 127, 889]] [890, [1, 2, 5, 10, 89, 178, 445, 890]] [891, [1, 3, 9, 11, 27, 33, 81, 99, 297, 891]] [892, [1, 2, 4, 223, 446, 892]] [893, [1, 19, 47, 893]] [894, [1, 2, 3, 6, 149, 298, 447, 894]] [895, [1, 5, 179, 895]] [896, [1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 448, 896]] [897, [1, 3, 13, 23, 39, 69, 299, 897]] [898, [1, 2, 449, 898]] [899, [1, 29, 31, 899]] [900, [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900]] [901, [1, 17, 53, 901]] [902, [1, 2, 11, 22, 41, 82, 451, 902]] [903, [1, 3, 7, 21, 43, 129, 301, 903]] [904, [1, 2, 4, 8, 113, 226, 452, 904]] [905, [1, 5, 181, 905]] [906, [1, 2, 3, 6, 151, 302, 453, 906]] [907, [1, 907]] [908, [1, 2, 4, 227, 454, 908]] [909, [1, 3, 9, 101, 303, 909]] [910, [1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910]] [911, [1, 911]] [912, [1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912]] [913, [1, 11, 83, 913]] [914, [1, 2, 457, 914]] [[915, [1, 3, 5, 15, 61, 183, 305, 915]] [916, [1, 2, 4, 229, 458, 916]] [917, [1, 7, 131, 917]] [918, [1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, 918]] [919, [1, 919]] [920, [1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920]] [921, [1, 3, 307, 921]] [922, [1, 2, 461, 922]] [923, [1, 13, 71, 923]] [924, [1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924]] [925, [1, 5, 25, 37, 185, 925]] [926, [1, 2, 463, 926]] [927, [1, 3, 9, 103, 309, 927]] [928, [1, 2, 4, 8, 16, 29, 32, 58, 116, 232, 464, 928]] [929, [1, 929]] [930, [1, 2, 3, 5, 6, 10, 15, 30, 31, 62, 93, 155, 186, 310, 465, 930]] [931, [1, 7, 19, 49, 133, 931]] [932, [1, 2, 4, 233, 466, 932]] [933, [1, 3, 311, 933]] [934, [1, 2, 467, 934]] [935, [1, 5, 11, 17, 55, 85, 187, 935]] [936, [1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 36, 39, 52, 72, 78, 104, 117, 156, 234, 312, 468, 936]] [937, [1, 937]] [938, [1, 2, 7, 14, 67, 134, 469, 938]] [939, [1, 3, 313, 939]] [940, [1, 2, 4, 5, 10, 20, 47, 94, 188, 235, 470, 940]] [941, [1, 941]] [942, [1, 2, 3, 6, 157, 314, 471, 942]] [943, [1, 23, 41, 943]] [944, [1, 2, 4, 8, 16, 59, 118, 236, 472, 944]] [945, [1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945]] [946, [1, 2, 11, 22, 43, 86, 473, 946]] [947, [1, 947]] [948, [1, 2, 3, 4, 6, 12, 79, 158, 237, 316, 474, 948]] [949, [1, 13, 73, 949]] [950, [1, 2, 5, 10, 19, 25, 38, 50, 95, 190, 475, 950]] [951, [1, 3, 317, 951]] [952, [1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952]] [953, [1, 953]] [954, [1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954]] [955, [1, 5, 191, 955]] [956, [1, 2, 4, 239, 478, 956]] [957, [1, 3, 11, 29, 33, 87, 319, 957]] [958, [1, 2, 479, 958]] [959, [1, 7, 137, 959]] [960, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960]] [961, [1, 31, 961]] [962, [1, 2, 13, 26, 37, 74, 481, 962]] [963, [1, 3, 9, 107, 321, 963]] [964, [1, 2, 4, 241, 482, 964]] [965, [1, 5, 193, 965]] [966, [1, 2, 3, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966]] [967, [1, 967]] [968, [1, 2, 4, 8, 11, 22, 44, 88, 121, 242, 484, 968]] [969, [1, 3, 17, 19, 51, 57, 323, 969]] [970, [1, 2, 5, 10, 97, 194, 485, 970]] [971, [1, 971]] [972, [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972]] [973, [1, 7, 139, 973]] [974, [1, 2, 487, 974]] [975, [1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975]] [976, [1, 2, 4, 8, 16, 61, 122, 244, 488, 976]] [977, [1, 977]] [978, [1, 2, 3, 6, 163, 326, 489, 978]] [979, [1, 11, 89, 979]] [980, [1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980]] [981, [1, 3, 9, 109, 327, 981]] [982, [1, 2, 491, 982]] [983, [1, 983]] [984, [1, 2, 3, 4, 6, 8, 12, 24, 41, 82, 123, 164, 246, 328, 492, 984]] [985, [1, 5, 197, 985]] [986, [1, 2, 17, 29, 34, 58, 493, 986]] [987, [1, 3, 7, 21, 47, 141, 329, 987]] [988, [1, 2, 4, 13, 19, 26, 38, 52, 76, 247, 494, 988]] [989, [1, 23, 43, 989]] [990, [1, 2, 3, 5, 6, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990]] [991, [1, 991]] [992, [1, 2, 4, 8, 16, 31, 32, 62, 124, 248, 496, 992]] [993, [1, 3, 331, 993]] [994, [1, 2, 7, 14, 71, 142, 497, 994]] [995, [1, 5, 199, 995]] [996, [1, 2, 3, 4, 6, 12, 83, 166, 249, 332, 498, 996]] [997, [1, 997]] [998, [1, 2, 499, 998]] [999, [1, 3, 9, 27, 37, 111, 333, 999]] [1000, [1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000]]].

Теперь нескладно обчислити і врахувати суму делителей чисел від 1 до 1000(которые теж отримано з допомогою програми Derive (за такою формулою 2.), тепер делители «a» просто складалися): [1, 1] [2, 3] [3, 4] [4, 7] [5, 6] [6, 12] [7, 8] [8, 15] [9, 13] [10, 18] [11, 12] [12, 28] [13, 14] [14, 24] [15, 24] [16, 31] [17, 18] [18, 39] [19, 20] [20, 42] [21, 32] [22, 36] [23, 24] [24, 60] [25, 31] [26, 42] [27, 40] [28, 56] [29, 30] [30, 72] [31, 32] [32, 63] [33, 48] [34, 54] [35, 48] [36, 91] [37, 38] [38, 60] [39, 56] [40, 90] [41, 42] [42, 96] [43, 44] [44, 84] [45, 78] [46, 72] [47, 48] [48, 124] [49, 57] [50, 93] [51, 72] [52, 98] [53, 54] [54, 120] [55, 72] [56, 120] [57, 80] [58, 90] [59, 60] [60, 168] [61, 62] [62, 96] [63, 104] [64, 127] [65, 84] [66, 144] [67, 68] [68, 126] [69, 96] [70, 144] [71, 72] [72, 195] [73, 74] [74, 114] [75, 124] [76, 140] [77, 96] [78, 168] [79, 80] [80, 186] [81, 121] [82, 126] [83, 84] [84, 224] [85, 108] [86, 132] [87, 120] [88, 180] [89, 90] [90, 234] [91, 112] [92, 168] [93, 128] [94, 144] [95, 120] [96, 252] [97, 98] [98, 171] [99, 156] [100, 217] [101, 102] [102, 216] [103, 104] [104, 210] [105, 192] [106, 162] [107, 108] [108, 280] [109, 110] [110, 216] [111, 152] [112, 248] [113, 114] [114, 240] [115, 144] [116, 210] [117, 182] [118, 180] [119, 144] [120, 360] [121, 133] [122, 186] [123, 168] [124, 224] [125, 156] [126, 312] [127, 128] [128, 255] [129, 176] [130, 252] [131, 132] [132, 336] [133, 160] [134, 204] [135, 240] [136, 270] [137, 138] [138, 288] [139, 140] [140, 336] [141, 192] [142, 216] [143, 168] [144, 403] [145, 180] [146, 222] [147, 228] [148, 266] [149, 150] [150, 372] [151, 152] [152, 300] [153, 234] [154, 288] [155, 192] [156, 392] [157, 158] [158, 240] [159, 216] [160, 378] [161, 192] [162, 363] [163, 164] [164, 294] [165, 288] [166, 252] [167, 168] [168, 480] [169, 183] [170, 324] [171, 260] [172, 308] [173, 174] [174, 360] [175, 248] [176, 372] [177, 240] [178, 270] [179, 180] [180, 546] [181, 182] [182, 336] [183, 248] [184, 360] [185, 228] [186, 384] [187, 216] [188, 336] [189, 320] [190, 360] [191, 192] [192, 508] [193, 194] [194, 294] [195, 336] [196, 399] [197, 198] [198, 468] [199, 200] [200, 465] [201, 272] [202, 306] [203, 240] [204, 504] [205, 252] [206, 312] [207, 312] [208, 434] [209, 240] [210, 576] [211, 212] [212, 378] [213, 288] [214, 324] [215, 264] [216, 600] [217, 256] [218, 330] [219, 296] [220, 504] [221, 252] [222, 456] [223, 224] [224, 504] [225, 403] [226, 342] [227, 228] [228, 560] [229, 230] [230, 432] [231, 384] [232, 450] [233, 234] [234, 546] [235, 288] [236, 420] [237, 320] [238, 432] [239, 240] [240, 744] [241, 242] [242, 399] [243, 364] [244, 434] [245, 342] [246, 504] [247, 280] [248, 480] [249, 336] [250, 468] [251, 252] [252, 728] [253, 288] [254, 384] [255, 432] [256, 511] [257, 258] [258, 528] [259, 304] [260, 588] [261, 390] [262, 396] [263, 264] [264, 720] [265, 324] [266, 480] [267, 360] [268, 476] [269, 270] [270, 720] [271, 272] [272, 558] [273, 448] [274, 414] [275, 372] [276, 672] [277, 278] [278, 420] [279, 416] [280, 720] [281, 282] [282, 576] [283, 284] [284, 504] [285, 480] [286, 504] [287, 336] [288, 819] [289, 307] [290, 540] [291, 392] [292, 518] [293, 294] [294, 684] [295, 360] [296, 570] [297, 480] [298, 450] [299, 336] [300, 868] [301, 352] [302, 456] [303, 408] [304, 620] [305, 372] [306, 702] [307, 308] [308, 672] [309, 416] [310, 576] [311, 312] [312, 840] [313, 314] [314, 474] [315, 624] [316, 560] [317, 318] [318, 648] [319, 360] [320, 762] [321, 432] [322, 576] [323, 360] [324, 847] [325, 434] [326, 492] [327, 440] [328, 630] [329, 384] [330, 864] [331, 332] [332, 588] [333, 494] [334, 504] [335, 408] [336, 992] [337, 338] [338, 549] [339, 456] [340, 756] [341, 384] [342, 780] [343, 400] [344, 660] [345, 576] [346, 522] [347, 348] [348, 840] [349, 350] [350, 744] [351, 560] [352, 756] [353, 354] [354, 720] [355, 432] [356, 630] [357, 576] [358, 540] [359, 360] [360, 1170] [361, 381] [362, 546] [363, 532] [364, 784] [365, 444] [366, 744] [367, 368] [368, 744] [369, 546] [370, 684] [371, 432] [372, 896] [373, 374] [374, 648] [375, 624] [376, 720] [377, 420] [378, 960] [379, 380] [380, 840] [381, 512] [382, 576] [383, 384] [384, 1020] [385, 576] [386, 582] [387, 572] [388, 686] [389, 390] [390, 1008] [391, 432] [392, 855] [393, 528] [394, 594] [395, 480] [396, 1092] [397, 398] [398, 600] [399, 640] [400, 961] [401, 402] [402, 816] [403, 448] [404, 714] [405, 726] [406, 720] [407, 456] [408, 1080] [409, 410] [410, 756] [411, 552] [412, 728] [413, 480] [414, 936] [415, 504] [416, 882] [417, 560] [418, 720] [419, 420] [420, 1344] [421, 422] [422, 636] [423, 624] [424, 810] [425, 558] [426, 864] [427, 496] [428, 756] [429, 672] [430, 792] [431, 432] [432, 1240] [433, 434] [434, 768] [435, 720] [436, 770] [437, 480] [438, 888] [439, 440] [440, 1080] [441, 741] [442, 756] [443, 444] [444, 1064] [445, 540] [446, 672] [447, 600] [448, 1016] [449, 450] [450, 1209] [451, 504] [452, 798] [453, 608] [454, 684] [455, 672] [456, 1200] [457, 458] [458, 690] [459, 720] [460, 1008] [461, 462] [462, 1152] [463, 464] [464, 930] [465, 768] [466, 702] [467, 468] [468, 1274] [469, 544] [470, 864] [471, 632] [472, 900] [473, 528] [474, 960] [475, 620] [476, 1008] [477, 702] [478, 720] [479, 480] [480, 1512] [481, 532] [482, 726] [483, 768] [484, 931] [485, 588] [486, 1092] [487, 488] [488, 930] [489, 656] [490, 1026] [491, 492] [492, 1176] [493, 540] [494, 840] [495, 936] [496, 992] [497, 576] [498, 1008] [499, 500] [500, 1092] [501, 672] [502, 756] [503, 504] [504, 1560] [505, 612] [506, 864] [507, 732] [508, 896] [509, 510] [510, 1296] [511, 592] [512, 1023] [513, 800] [514, 774] [515, 624] [516, 1232] [517, 576] [518, 912] [519, 696] [520, 1260] [521, 522] [522, 1170] [523, 524] [524, 924] [525, 992] [526, 792] [527, 576] [528, 1488] [529, 553] [530, 972] [531, 780] [532, 1120] [533, 588] [534, 1080] [535, 648] [536, 1020] [537, 720] [538, 810] [539, 684] [540, 1680] [541, 542] [542, 816] [543, 728] [544, 1134] [545, 660] [546, 1344] [547, 548] [548, 966] [549, 806] [550, 1116] [551, 600] [552, 1440] [553, 640] [554, 834] [555, 912] [556, 980] [557, 558] [558, 1248] [559, 616] [560, 1488] [561, 864] [562, 846] [563, 564] [564, 1344] [565, 684] [566, 852] [567, 968] [568, 1080] [569, 570] [570, 1440] [571, 572] [572, 1176] [573, 768] [574, 1008] [575, 744] [576, 1651] [577, 578] [578, 921] [579, 776] [580, 1260] [581, 672] [582, 1176] [583, 648] [584, 1110] [585, 1092] [586, 882] [587, 588] [588, 1596] [589, 640] [590, 1080] [591, 792] [592, 1178] [593, 594] [594, 1440] [595, 864] [596, 1050] [597, 800] [598, 1008] [599, 600] [600, 1860] [601, 602] [602, 1056] [603, 884] [604, 1064] [605, 798] [606, 1224] [607, 608] [608, 1260] [609, 960] [610, 1116] [611, 672] [612, 1638] [613, 614] [614, 924] [615, 1008] [616, 1440] [617, 618] [618, 1248] [619, 620] [620, 1344] [621, 960] [622, 936] [623, 720] [624, 1736] [625, 781] [626, 942] [627, 960] [628, 1106] [629, 684] [630, 1872] [631, 632] [632, 1200] [633, 848] [634, 954] [635, 768] [636, 1512] [637, 798] [638, 1080] [639, 936] [640, 1530] [641, 642] [642, 1296] [643, 644] [644, 1344] [645, 1056] [646, 1080] [647, 648] [648, 1815] [649, 720] [650, 1302] [651, 1024] [652, 1148] [653, 654] [654, 1320] [655, 792] [656, 1302] [657, 962] [658, 1152] [659, 660] [660, 2016] [661, 662] [662, 996] [663, 1008] [664, 1260] [665, 960] [666, 1482] [667, 720] [668, 1176] [669, 896] [670, 1224] [671, 744] [672, 2016] [673, 674] [674, 1014] [675, 1240] [676, 1281] [677, 678] [678, 1368] [679, 784] [680, 1620] [681, 912] [682, 1152] [683, 684] [684, 1820] [685, 828] [686, 1200] [687, 920] [688, 1364] [689, 756] [690, 1728] [691, 692] [692, 1218] [693, 1248] [694, 1044] [695, 840] [696, 1800] [697, 756] [698, 1050] [699, 936] [700, 1736] [701, 702] [702, 1680] [703, 760] [704, 1524] [705, 1152] [706, 1062] [707, 816] [708, 1680] [709, 710] [710, 1296] [711, 1040] [712, 1350] [713, 768] [714, 1728] [715, 1008] [716, 1260] [717, 960] [718, 1080] [719, 720] [720, 2418] [721, 832] [722, 1143] [723, 968] [724, 1274] [725, 930] [726, 1596] [727, 728] [728, 1680] [729, 1093] [730, 1332] [731, 792] [732, 1736] [733, 734] [734, 1104] [735, 1368] [736, 1512] [737, 816] [738, 1638] [739, 740] [740, 1596] [741, 1120] [742, 1296] [743, 744] [744, 1920] [745, 900] [746, 1122] [747, 1092] [748, 1512] [749, 864] [750, 1872] [751, 752] [752, 1488] [753, 1008] [754, 1260] [755, 912] [756, 2240] [757, 758] [758, 1140] [759, 1152] [760, 1800] [761, 762] [762, 1536] [763, 880] [764, 1344] [765, 1404] [766, 1152] [767, 840] [768, 2044] [769, 770] [770, 1728] [771, 1032] [772, 1358] [773, 774] [774, 1716] [775, 992] [776, 1470] [777, 1216] [778, 1170] [779, 840] [780, 2352] [781, 864] [782, 1296] [783, 1200] [784, 1767] [785, 948] [786, 1584] [787, 788] [788, 1386] [789, 1056] [790, 1440] [791, 912] [792, 2340] [793, 868] [794, 1194] [795, 1296] [796, 1400] [797, 798] [798, 1920] [799, 864] [800, 1953] [801, 1170] [802, 1206] [803, 888] [804, 1904] [805, 1152] [806, 1344] [807, 1080] [808, 1530] [809, 810] [810, 2178] [811, 812] [812, 1680] [813, 1088] [814, 1368] [815, 984] [816, 2232] [817, 880] [818, 1230] [819, 1456] [820, 1764] [821, 822] [822, 1656] [823, 824] [824, 1560] [825, 1488] [826, 1440] [827, 828] [828, 2184] [829, 830] [830, 1512] [831, 1112] [832, 1778] [833, 1026] [834, 1680] [835, 1008] [836, 1680] [837, 1280] [838, 1260] [839, 840] [840, 2880] [841, 871] [842, 1266] [843, 1128] [844, 1484] [845, 1098] [846, 1872] [847, 1064] [848, 1674] [849, 1136] [850, 1674] [851, 912] [852, 2016] [853, 854] [854, 1488] [855, 1560] [856, 1620] [857, 858] [858, 2016] [859, 860] [860, 1848] [861, 1344] [862, 1296] [863, 864] [864, 2520] [865, 1044] [866, 1302] [867, 1228] [868, 1792] [869, 960] [870, 2160] [871, 952] [872, 1650] [873, 1274] [874, 1440] [875, 1248] [876, 2072] [877, 878] [878, 1320] [879, 1176] [880, 2232] [881, 882] [882, 2223] [883, 884] [884, 1764] [885, 1440] [886, 1332] [887, 888] [888, 2280] [889, 1024] [890, 1620] [891, 1452] [892, 1568] [893, 960] [894, 1800] [895, 1080] [896, 2040] [897, 1344] [898, 1350] [899, 960] [900, 2821] [901, 972] [902, 1512] [903, 1408] [904, 1710] [905, 1092] [906, 1824] [907, 908] [908, 1596] [909, 1326] [910, 2016] [911, 912] [912, 2480] [913, 1008] [914, 1374] [915, 1488] [916, 1610] [917, 1056] [918, 2160] [919, 920] [920, 2160] [921, 1232] [922, 1386] [923, 1008] [924, 2688] [925, 1178] [926, 1392] [927, 1352] [928, 1890] [929, 930] [930, 2304] [931, 1140] [932, 1638] [933, 1248] [934, 1404] [935, 1296] [936, 2730] [937, 938] [938, 1632] [939, 1256] [940, 2016] [941, 942] [942, 1896] [943, 1008] [944, 1860] [945, 1920] [946, 1584] [947, 948] [948, 2240] [949, 1036] [950, 1860] [951, 1272] [952, 2160] [953, 954] [954, 2106] [955, 1152] [956, 1680] [957, 1440] [958, 1440] [959, 1104] [960, 3048] [961, 993] [962, 1596] [963, 1404] [964, 1694] [965, 1164] [966, 2304] [967, 968] [968, 1995] [969, 1440] [970, 1764] [971, 972] [972, 2548] [973, 1120] [974, 1464] [975, 1736] [976, 1922] [977, 978] [978, 1968] [979, 1080] [980, 2394] [981, 1430] [982, 1476] [983, 984] [984, 2520] [985, 1188] [986, 1620] [987, 1536] [988, 1960] [989, 1056] [990, 2808] [991, 992] [992, 2016] [993, 1328] [994, 1728] [995, 1200] [996, 2352] [997, 998] [998, 1500] [999, 1520] [1000, 2340] Тепер подивімося, чи всі числа є сумою делителей будь-якого числа і чи є такі числа сума делителей яких дорівнює (у перших двох сотнях). Нижче приведено таблиця: [[4, 7]](на місці сума делителей, але в першому число з цією сумою делителей) … [[1, 1]], [2] (тобто. немає такої числа з сумою делителей рівної двум):

[1,1].

[2].

[2,3].

[3,4].

[5].

[5,6].

[4,7].

[7,8].

[9].

[10].

[11].

[6,12].

[11, 12].

[9,13].

[13,14].

[8,15].

[16].

[17].

[10,18].

[17,18].

[19].

[19.20].

[21].

[22].

[23].

[14,24].

[15,24].

[23,24].

[25].

[26].

[27].

[12, 28].

[29].

[29,30].

[16,31].

[25.31].

[21,32].

[31,32].

[33].

[34].

[35].

[22,36].

[37].

[37,38].

[18,39].

[27, 40].

[41].

[20,42].

[26,42].

[41,42].

[43].

[43,44].

[45].

[46].

[47].

[33,48].

[35,4 8].

[47,48].

[49].

[50].

[51].

[52].

[53].

[34,54].

[53, 54].

[55].

[28,56].

[39.56].

[49,57].

[58].

[59].

[24,60].

[38.60].

[59,60].

[61].

[61,62].

[32,63].

[64].

[65].

[66].

[67].

[67, 68].

[69].

[70].

[71].

[30,72].

[46,72].

[51,72].

[55,72].

[71,72].

[73].

[73,74].

[75].

[76].

[77].

[45,78].

[79].

[57,80].

[79,80].

[81].

[82].

[83].

[44,84].

[65,84].

[83,84].

[85].

[86].

[87].

[88].

[89].

[40, 90].

[58,90].

[89,90].

[36,91].

[92].

[50,93].

[94].

[95].

[42, 96].

[62,96].

[69,96].

[77,96].

[97].

[52,98].

[97,98].

[99].

[100].

[101].

[102].

[103].

[63,104].

[105].

[106].

[107].

[85,108].

[109].

[110].

[111].

[91, 112].

[113].

[74,114],.

[115].

[116].

[117].

[118].

[119].

[54,120].

[56,120].

[87,120].

[95,120].

[81,121].

[122].

[123].

[48,124].

[75, 124].

[125].

[68,126].

[82.126].

[64,127].

[9 3,128].

[129].

[130].

[131].

[86,132].

[133].

[134].

[135].

[136].

[137].

[138].

[139].

[76,140].

[141].

[142].

[143].

[66,144].

[70,144].

[94,144].

[145].

[146].

[147].

[178].

[149].

[150].

[151].

[152].

[153].

[154].

[155].

[99,156].

[157].

[158].

[159].

[160].

[161].

[162].

[163].

[164].

[165].

[166].

[167].

[60,168].

[78,168].

[92,168].

[169].

[170].

[98,171].

[172].

[173].

[174].

[175].

[176].

[177].

[178].

[179].

[88,180].

[181].

[182].

[183].

[184].

[185].

[80,186].

[187].

[188].

[189].

[190].

[191].

[192].

[193].

[194].

[72,195].

[196].

[197].

[198].

[199].

[200] Як помітили, є такі числа, які є сумою делителей ні одного числа і такий самий є такі числа, що є сумою делителей жодного, а кількох чисел. Тепер подивімося ті числа, які є сумою делителей жодного, а кількох чисел: [6,12], [11,12] [10,18], [17,18] [14,24], [15,24], [23,24] [16,31]. [25,31] [21,32], [31,32] [20, 42], [26,42], [41,42] [33,48], [35,48], [47,48] [34,5 4], [53,54] [28,56], [39,56] [24,60], [38,60], [59, 60] [30,72], [46,72], [51,72], [55,72], [71,72] [57,80], [79,80] [44,84], [65,84], [83,84] [40,90], [58, 9 0], [89,90] [42,96], [62,96], [69,96], [77,96] [52,98], [97,98] [54,120], [56, 120], [87,120], [95,120] [48,124], [75,124] [68,126], [82,126] [66,144], [70, 144], [94,144] [60,168], [78,168], [92,168].

Звідси можна дійти невтішного висновку, що перебування числа з його сумі делителей який завжди можливо, й який завжди однозначно.

Тепер побудуємо графік. За віссю Х розташуємо числа, а, по осі Y їх суму делителей (числа від 1 до 1000): Подивимося, що саме ми вийшло: на графіці чітко проглядаються кілька прямих ліній, наприклад, нижня це — прості числа. Верхня кордон — це найскладніші числа (які мають найбільший кількість делителей) — це пряма, але й парабола. Найімовірніше, — це показова функція (у = ах).

У мемуарах Эйлера я знайшов багато цікавих мені рассуждений (?(n) — сума делителей числа n): Визначивши значення ?(n) ми ясно бачимо, що й p — просте, то ?(p)= p + 1. ?(1)=1, і якщо число n — складене, то ?(n)>1 + n.

Якщо a, b, з, d — різні прості числа, ми видим:

?(ab)=1+a+b+ab=(1+a)(1+b)= ?(a)?(b).

?(abcd)= ?(a)?(b)?©?(d).

?(a²)=1+a+a2=[pic].

?(a³)=1+a+a2+a3=[pic].

І вообще.

?(nn)=[pic].

Користуючись этим:

?(aqbwcedr)= ?(aq)?(bw)?(ce)?(dr).

Наприклад ?(360), 360 = 23*32*5 => ?(23) ?(32) ?(5)=15*13*6=1170.

Щоб показати послідовність сум делителей наведемо таблицу:

|n |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 | |0 |- |1 |3 |4 |7 |6 |12 |8 |15 |13 | |10 |18 |12 |28 |14 |24 |24 |31 |18 |39 |20 | |20 |42 |32 |36 |24 |60 |31 |42 |40 |56 |30 | |30 |72 |32 |63 |48 |54 |48 |91 |38 |60 |56 | |40 |90 |42 |96 |44 |84 |78 |72 |48 |124 |57 | |50 |93 |72 |98 |54 |120 |72 |120 |80 |90 |60 | |60 |168 |62 |96 |104 |127 |84 |144 |68 |126 |96 | |70 |144 |72 |195 |74 |114 |424 |140 |96 |168 |80 | |80 |186 |121 |126 |84 |224 |108 |132 |120 |180 |90 | |90 |234 |112 |168 |128 |144 |120 |252 |98 |171 |156 |.

Якщо ?(n) позначає член будь-який цієї послідовності, а ?(n — 1), ?(n — 2), ?(n — 3)… попередні члени, то ?(n) можна отримати по кільком попереднім членам:

?(n) = ?(n — 1) + ?(n — 2) — ?(n — 5) — ?(n — 7) + ?(n — 12) + ?(n — 15) — ?(n — 22) — ?(n — 26) + … (**).

Знаки «+» «-» у правій частині формули попарно чергуються. Закон чисел 1, 2, 5, 7, 12, 15…, которые ми повинні вичитати з аналізованого числа n, стане ясний якщо взяти їх разности:

Числа:1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100… Різниці: 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8…

У насправді, маємо тут по черзі все цілі числа 1, 2, 3, 4, 5, 6, 7… і непарні 3, 5, 7,9 11…

Хоча цей послідовність нескінченна, ми мають у кожній оказії брати ті члени, котрим числа які стоять під знаком ?, ще позитивні, і опускати? для негативних чисел. Якщо нашої формулі зустрітися ?(0), то, оскільки його значення саме собі є неопределённым, ми повинні підставити замість ?(0) аналізованих число n. Примеры:

?(1) = ?(0) =1.

= 1 ?(2) = ?(1) + ?(0) = 1 + 2.

= 3 … ?(20) = ?(19)+?(18)-?(15)-?(13)+9?(8)+?(5)=20+39−24−14+15+6= 42.

Доказ теореми (**) я приводити не буду.

Взагалі, знайти суму всіх делителей числа можна з допомогою канонічного розкладання натурального числа (це вже зазначалося). Суму делителей числа n позначають ?(n). Легко знайти ?(n) для невеликих натуральних чисел, наприклад ?(12) = 1+2+3+4+6+12=28(это привели вище). Але у досить великих числах відшукування всіх делителей, а тим паче суми стає важким. Зовсім інша річ, коли вже відомо, що канонічне розкладання числа n таково:[pic]. Його делителями є всі числа [pic], котрим 0? ?p.s? ?p.s, p. s = 1, …, k. Зрозуміло, що ?(n) є сумою всіх таких чисел що за різних значеннях показників ?1, ?2, … ?k. Цей результат ми матимемо розкривши дужки в произведении.

[pic] За формулою кінцевого числа членів геометричній прогресії дійшли равенству.

[pic] (*).

По цієї формули ?(360) = [pic] .

Формулу для обчислення значення функції ?(n) вивів чудовий англійський математик Джон Валлис (1616 — 1703) — одного з засновників і перших членів Лондонського Королівства суспільства (Академії наук). Він був із англійських математиків, почали займатися математичним аналізом. Йому належать багато позначення і терміни, застосовувані нині у математиці, зокрема знак? для позначення нескінченності. Валлис вивів дивовижну формулу, представляє число? як нескінченного произведения:

[pic].

Д. Валлис багато займався комбінаторикою і її додатками до теорії шифрів, не безпідставно вважаючи себе родоначальником нової науки — криптологии (від грецьк. «криптос» — таємний, «логос» — наука, вчення). Він був однією з кращих шифрувальників свого часу й за дорученням міністра поліції Терло навчався республіканському уряді Кромвеля розшифровуванням послань монархічних заговорщиков.

З функцією ?(n) пов’язаний ряд цікавих завдань. Например:

1.) Знайти пару цілих чисел, які відповідають умові: ?(m1)=m2, ?(m2)=m1.

Деякі їх не вдається вирішити нам навіть з допомогою формули (*). Приміром, не інакше як добором можна знайти числа, котрим ?(n) є квадрат деякого натурального числа. Такими числами є 22, 66, 70, 81, 343, 1501, 4 479 865. Ось ще два завдання, приведені в 1657 р. П'єром Ферма: 1. знайти таке m, котрій ?(m3) — квадрат натурального числа (Ферма знайшов не одне правильне рішення це завдання); 2. знайти таке m, котрій ?(m2) — куб натурального числа. Наприклад, однією з рішень першого завдання є m = 7, а другий m = 43 098. З допомогою програми Derive, я спробував знайти ще рішення і маю якесь нічого немає. (я розглядав ?(m3) = n2, де m приймає значення від 1 до 1000, а n від 1 до 5000 один.) і також саме 2.)).

Формули: 1. DELITELI (m) := SELECT (MOD (m, n) = 0, n, 1, m).

DIMENSION (DELITELI (m)).

2. SUMMADELITELEY (m) :=? ELEMENT (DELITELI (m), і) i=1.