Научная Петербургская Академия

Сумма делителей числа - (реферат)

Сумма делителей числа - (реферат)

Дата добавления: март 2006г.

    Сумма делителей числа

Для начало приведём экспериментальный материал (который был получен с помощью программы Derive (по формуле 1. (см. ниже)): для нахождения делителей числа “a”, программа делила число “a” на другие числа не превосходящие само число и если остаток от деления был равен 0, то число записывалось как делитель “a”. ): Ниже приведены все делители чисел от 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]]
    [467, [1, 467]]

[468, [1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468]] [469, [1, 7, 67, 469]]

    [470, [1, 2, 5, 10, 47, 94, 235, 470]]
    [471, [1, 3, 157, 471]]
    [472, [1, 2, 4, 8, 59, 118, 236, 472]]
    [473, [1, 11, 43, 473]]
    [474, [1, 2, 3, 6, 79, 158, 237, 474]]
    [475, [1, 5, 19, 25, 95, 475]]
    [476, [1, 2, 4, 7, 14, 17, 28, 34, 68, 119, 238, 476]]
    [477, [1, 3, 9, 53, 159, 477]]
    [478, [1, 2, 239, 478]]
    [479, [1, 479]]

[480, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, 480]]

    [481, [1, 13, 37, 481]]
    [482, [1, 2, 241, 482]]
    [483, [1, 3, 7, 21, 23, 69, 161, 483]]
    [484, [1, 2, 4, 11, 22, 44, 121, 242, 484]]
    [485, [1, 5, 97, 485]]
    [486, [1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486]]
    [487, [1, 487]]
    [488, [1, 2, 4, 8, 61, 122, 244, 488]]
    [489, [1, 3, 163, 489]]
    [490, [1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]]
    [491, [1, 491]]
    [492, [1, 2, 3, 4, 6, 12, 41, 82, 123, 164, 246, 492]]
    [493, [1, 17, 29, 493]]
    [494, [1, 2, 13, 19, 26, 38, 247, 494]]
    [495, [1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495]]
    [496, [1, 2, 4, 8, 16, 31, 62, 124, 248, 496]]
    [497, [1, 7, 71, 497]]
    [498, [1, 2, 3, 6, 83, 166, 249, 498]]
    [499, [1, 499]]
    [500, [1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500]]
    [501, [1, 3, 167, 501]]
    [502, [1, 2, 251, 502]]
    [503, [1, 503]]

[504, [1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504]]

    [505, [1, 5, 101, 505]]
    [506, [1, 2, 11, 22, 23, 46, 253, 506]]
    [507, [1, 3, 13, 39, 169, 507]]
    [508, [1, 2, 4, 127, 254, 508]]
    [509, [1, 509]]

[510, [1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, 510]] [511, [1, 7, 73, 511]]

    [512, [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]]
    [513, [1, 3, 9, 19, 27, 57, 171, 513]]
    [514, [1, 2, 257, 514]]
    [515, [1, 5, 103, 515]]
    [516, [1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516]]
    [517, [1, 11, 47, 517]]
    [518, [1, 2, 7, 14, 37, 74, 259, 518]]
    [519, [1, 3, 173, 519]]

[520, [1, 2, 4, 5, 8, 10, 13, 20, 26, 40, 52, 65, 104, 130, 260, 520]] [521, [1, 521]]

    [522, [1, 2, 3, 6, 9, 18, 29, 58, 87, 174, 261, 522]]
    [523, [1, 523]]
    [524, [1, 2, 4, 131, 262, 524]]
    [525, [1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525]]
    [526, [1, 2, 263, 526]]
    [527, [1, 17, 31, 527]]

[528, [1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528]]

    [529, [1, 23, 529]]
    [530, [1, 2, 5, 10, 53, 106, 265, 530]]
    [531, [1, 3, 9, 59, 177, 531]]
    [532, [1, 2, 4, 7, 14, 19, 28, 38, 76, 133, 266, 532]]
    [533, [1, 13, 41, 533]]
    [534, [1, 2, 3, 6, 89, 178, 267, 534]]
    [535, [1, 5, 107, 535]]
    [536, [1, 2, 4, 8, 67, 134, 268, 536]]
    [537, [1, 3, 179, 537]]
    [538, [1, 2, 269, 538]]
    [539, [1, 7, 11, 49, 77, 539]]

[540, [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540]]

    [541, [1, 541]]
    [542, [1, 2, 271, 542]]
    [543, [1, 3, 181, 543]]
    [544, [1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, 544]]
    [545, [1, 5, 109, 545]]

[546, [1, 2, 3, 6, 7, 13, 14, 21, 26, 39, 42, 78, 91, 182, 273, 546]] [547, [1, 547]]

    [548, [1, 2, 4, 137, 274, 548]]
    [549, [1, 3, 9, 61, 183, 549]]
    [550, [1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275, 550]]
    [551, [1, 19, 29, 551]]

[552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552]] [553, [1, 7, 79, 553]]

    [554, [1, 2, 277, 554]]
    [555, [1, 3, 5, 15, 37, 111, 185, 555]]
    [556, [1, 2, 4, 139, 278, 556]]
    [557, [1, 557]]
    [558, [1, 2, 3, 6, 9, 18, 31, 62, 93, 186, 279, 558]]
    [559, [1, 13, 43, 559]]

[560, [1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560]]

    [561, [1, 3, 11, 17, 33, 51, 187, 561]]
    [562, [1, 2, 281, 562]]
    [563, [1, 563]]
    [564, [1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564]]
    [565, [1, 5, 113, 565]]
    [566, [1, 2, 283, 566]]
    [567, [1, 3, 7, 9, 21, 27, 63, 81, 189, 567]]
    [568, [1, 2, 4, 8, 71, 142, 284, 568]]
    [569, [1, 569]]

[570, [1, 2, 3, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570]] [571, [1, 571]]

    [572, [1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572]]
    [573, [1, 3, 191, 573]]
    [574, [1, 2, 7, 14, 41, 82, 287, 574]]
    [575, [1, 5, 23, 25, 115, 575]]

[576, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576]]

    [577, [1, 577]]
    [578, [1, 2, 17, 34, 289, 578]]
    [579, [1, 3, 193, 579]]
    [580, [1, 2, 4, 5, 10, 20, 29, 58, 116, 145, 290, 580]]
    [581, [1, 7, 83, 581]]
    [582, [1, 2, 3, 6, 97, 194, 291, 582]]
    [583, [1, 11, 53, 583]]
    [584, [1, 2, 4, 8, 73, 146, 292, 584]]
    [585, [1, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585]]
    [586, [1, 2, 293, 586]]
    [587, [1, 587]]

[588, [1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588]] [589, [1, 19, 31, 589]]

    [590, [1, 2, 5, 10, 59, 118, 295, 590]]
    [591, [1, 3, 197, 591]]
    [592, [1, 2, 4, 8, 16, 37, 74, 148, 296, 592]]
    [593, [1, 593]]

[594, [1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594]] [595, [1, 5, 7, 17, 35, 85, 119, 595]]

    [596, [1, 2, 4, 149, 298, 596]]
    [597, [1, 3, 199, 597]]
    [598, [1, 2, 13, 23, 26, 46, 299, 598]]
    [599, [1, 599]]

[600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600]]

    [601, [1, 601]]
    [602, [1, 2, 7, 14, 43, 86, 301, 602]]
    [603, [1, 3, 9, 67, 201, 603]]
    [604, [1, 2, 4, 151, 302, 604]]
    [605, [1, 5, 11, 55, 121, 605]]
    [606, [1, 2, 3, 6, 101, 202, 303, 606]]
    [607, [1, 607]]
    [608, [1, 2, 4, 8, 16, 19, 32, 38, 76, 152, 304, 608]]
    [609, [1, 3, 7, 21, 29, 87, 203, 609]]
    [610, [1, 2, 5, 10, 61, 122, 305, 610]]
    [611, [1, 13, 47, 611]]

[612, [1, 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204, 306, 612]] [613, [1, 613]]

    [614, [1, 2, 307, 614]]
    [615, [1, 3, 5, 15, 41, 123, 205, 615]]

[616, [1, 2, 4, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616]] [617, [1, 617]]

    [618, [1, 2, 3, 6, 103, 206, 309, 618]]
    [619, [1, 619]]
    [620, [1, 2, 4, 5, 10, 20, 31, 62, 124, 155, 310, 620]]
    [621, [1, 3, 9, 23, 27, 69, 207, 621]]
    [622, [1, 2, 311, 622]]
    [623, [1, 7, 89, 623]]

[624, [1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624]]

    [625, [1, 5, 25, 125, 625]]
    [626, [1, 2, 313, 626]]
    [627, [1, 3, 11, 19, 33, 57, 209, 627]]
    [628, [1, 2, 4, 157, 314, 628]]
    [629, [1, 17, 37, 629]]

[630, [1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630]]

    [631, [1, 631]]
    [632, [1, 2, 4, 8, 79, 158, 316, 632]]
    [633, [1, 3, 211, 633]]
    [634, [1, 2, 317, 634]]
    [635, [1, 5, 127, 635]]
    [636, [1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636]]
    [637, [1, 7, 13, 49, 91, 637]]
    [638, [1, 2, 11, 22, 29, 58, 319, 638]]
    [639, [1, 3, 9, 71, 213, 639]]

[640, [1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640]] [641, [1, 641]]

    [642, [1, 2, 3, 6, 107, 214, 321, 642]]
    [643, [1, 643]]
    [644, [1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644]]
    [645, [1, 3, 5, 15, 43, 129, 215, 645]]
    [646, [1, 2, 17, 19, 34, 38, 323, 646]]
    [647, [1, 647]]

[648, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648]]

    [649, [1, 11, 59, 649]]
    [650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650]]
    [651, [1, 3, 7, 21, 31, 93, 217, 651]]
    [652, [1, 2, 4, 163, 326, 652]]
    [653, [1, 653]]
    [654, [1, 2, 3, 6, 109, 218, 327, 654]]
    [655, [1, 5, 131, 655]]
    [656, [1, 2, 4, 8, 16, 41, 82, 164, 328, 656]]
    [657, [1, 3, 9, 73, 219, 657]]
    [658, [1, 2, 7, 14, 47, 94, 329, 658]]
    [659, [1, 659]]

[660, [1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660]]

    [661, [1, 661]]
    [662, [1, 2, 331, 662]]
    [663, [1, 3, 13, 17, 39, 51, 221, 663]]
    [664, [1, 2, 4, 8, 83, 166, 332, 664]]
    [665, [1, 5, 7, 19, 35, 95, 133, 665]]
    [666, [1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666]]
    [667, [1, 23, 29, 667]]
    [668, [1, 2, 4, 167, 334, 668]]
    [669, [1, 3, 223, 669]]
    [670, [1, 2, 5, 10, 67, 134, 335, 670]]
    [671, [1, 11, 61, 671]]

[672, [1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672]]

    [673, [1, 673]]
    [674, [1, 2, 337, 674]]
    [675, [1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675]]
    [676, [1, 2, 4, 13, 26, 52, 169, 338, 676]]
    [677, [1, 677]]
    [678, [1, 2, 3, 6, 113, 226, 339, 678]]
    [679, [1, 7, 97, 679]]

[680, [1, 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, 340, 680]] [681, [1, 3, 227, 681]]

    [682, [1, 2, 11, 22, 31, 62, 341, 682]]
    [683, [1, 683]]

[684, [1, 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684]] [685, [1, 5, 137, 685]]

    [686, [1, 2, 7, 14, 49, 98, 343, 686]]
    [687, [1, 3, 229, 687]]
    [688, [1, 2, 4, 8, 16, 43, 86, 172, 344, 688]]
    [689, [1, 13, 53, 689]]

[690, [1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690]] [691, [1, 691]]

    [692, [1, 2, 4, 173, 346, 692]]
    [693, [1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693]]
    [694, [1, 2, 347, 694]]
    [695, [1, 5, 139, 695]]

[696, [1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, 696]] [697, [1, 17, 41, 697]]

    [698, [1, 2, 349, 698]]
    [699, [1, 3, 233, 699]]

[700, [1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700]] [701, [1, 701]]

[702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702]] [703, [1, 19, 37, 703]]

[704, [1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704]] [705, [1, 3, 5, 15, 47, 141, 235, 705]]

    [706, [1, 2, 353, 706]]
    [707, [1, 7, 101, 707]]
    [708, [1, 2, 3, 4, 6, 12, 59, 118, 177, 236, 354, 708]]
    [709, [1, 709]]
    [710, [1, 2, 5, 10, 71, 142, 355, 710]]
    [711, [1, 3, 9, 79, 237, 711]]
    [712, [1, 2, 4, 8, 89, 178, 356, 712]]
    [713, [1, 23, 31, 713]]

[714, [1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714]] [715, [1, 5, 11, 13, 55, 65, 143, 715]]

    [716, [1, 2, 4, 179, 358, 716]]
    [717, [1, 3, 239, 717]]
    [718, [1, 2, 359, 718]]
    [719, [1, 719]]

[720, [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720]]

    [721, [1, 7, 103, 721]]
    [722, [1, 2, 19, 38, 361, 722]]
    [723, [1, 3, 241, 723]]
    [724, [1, 2, 4, 181, 362, 724]]
    [725, [1, 5, 25, 29, 145, 725]]
    [726, [1, 2, 3, 6, 11, 22, 33, 66, 121, 242, 363, 726]]
    [727, [1, 727]]

[728, [1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, 728]] [729, [1, 3, 9, 27, 81, 243, 729]]

    [730, [1, 2, 5, 10, 73, 146, 365, 730]]
    [731, [1, 17, 43, 731]]
    [732, [1, 2, 3, 4, 6, 12, 61, 122, 183, 244, 366, 732]]
    [733, [1, 733]]
    [734, [1, 2, 367, 734]]
    [735, [1, 3, 5, 7, 15, 21, 35, 49, 105, 147, 245, 735]]
    [736, [1, 2, 4, 8, 16, 23, 32, 46, 92, 184, 368, 736]]
    [737, [1, 11, 67, 737]]
    [738, [1, 2, 3, 6, 9, 18, 41, 82, 123, 246, 369, 738]]
    [739, [1, 739]]
    [740, [1, 2, 4, 5, 10, 20, 37, 74, 148, 185, 370, 740]]
    [741, [1, 3, 13, 19, 39, 57, 247, 741]]
    [742, [1, 2, 7, 14, 53, 106, 371, 742]]
    [743, [1, 743]]

[744, [1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744]] [745, [1, 5, 149, 745]]

    [746, [1, 2, 373, 746]]
    [747, [1, 3, 9, 83, 249, 747]]
    [748, [1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748]]
    [749, [1, 7, 107, 749]]

[750, [1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750]] [751, [1, 751]]

    [752, [1, 2, 4, 8, 16, 47, 94, 188, 376, 752]]
    [753, [1, 3, 251, 753]]
    [754, [1, 2, 13, 26, 29, 58, 377, 754]]
    [755, [1, 5, 151, 755]]

[756, [1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 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 несложно посчитать и сумму делителей чисел от 1 до 1000(которые тоже были получены с помощью программы Derive (по формуле 2. ), теперь делители “a” просто складывались):

    [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]

Теперь посмотрим, все ли числа являются суммой делителей какого-либо числа и есть ли такие числа сумма делителей которых равна (в первых двух сотнях). Ниже приведена таблица: [[4, 7]](на втором месте сумма делителей, а на первом число с данной суммой делителей) … [[1, 1]], [2] (т. е. нет такого числа с суммой делителей равной 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]

Как мы заметили, есть такие числа, которые не являются суммой делителей ни одного числа и так же есть такие числа, которые являются суммой делителей ни одного, а нескольких чисел. Теперь посмотрим только те числа, которые являются суммой делителей ни одного, а нескольких 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]

Отсюда можно сделать вывод, что нахождение числа по его сумме делителей не всегда возможно и не всегда однозначно.

Теперь построим график. По оси Х расположим числа, а по оси Y их сумму делителей (числа от 1 до 1000):

Посмотрим, что же у нас получилось: на графике отчётливо просматриваются несколько прямых линий, например, нижняя это– простые числа. Верхняя граница –это наиболее сложные числа (имеющие наибольшее количество делителей) - это не прямая, но и не парабола. Скорее всего, – это показательная функция (у = ах). В мемуарах Эйлера я нашел много интересных мне рассуждений(у(n) – сумма делителей числа n): Определив значение у(n) мы ясно видим, что если p – простое, то у(p)= p + 1. у(1)=1, а если число n – составное, то у(n)>1 + n. Если a, b, c, d – различные простые числа, то мы видим:

    у(ab)=1+a+b+ab=(1+a)(1+b)= у(a)у(b)
    у(abcd)= у(a)у(b)у(c)у(d)
    у(a^2)=1+a+a2=
    у(a^3)=1+a+a2+a3=
    И вообще
    у(nn)=
    Пользуясь этим:
    у(aqbwcedr)= у(aq)у(bw)у(ce)у(dr)

Например у(360), 360 = 23*32*5 => у(23) у(32) у(5)=15*13*6=1170. Чтобы показать последовательность сумм делителей приведём 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

Если у(n) обозначает член любой этой последовательности, а у(n - 1), у(n - 2), у(n - 3)… предшествующие члены, то у(n) всегда можно получить по нескольким предыдущим членам:

у(n) = у(n - 1) + у(n - 2) - у(n - 5) - у(n - 7) + у(n - 12) + у(n - 15) - у(n - 22) - у(n– 26) + … (**)

Знаки “+” “-” в правой части формулы попарно чередуются. Закон чисел 1, 2, 5, 7, 12, 15…, которые мы должны вычитать из рассматриваемого числа n, станет ясен если мы возьмем их разности:

Числа: 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100… Разности: 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8… В самом деле, мы имеем здесь поочередно все целые числа 1, 2, 3, 4, 5, 6, 7… и нечетные 3, 5, 7, 9 11…

Хотя эта последовательность бесконечна, мы должны в каждом случае брать только те члены, для которых числа стоящие под знаком у, еще положительны, и опускать у для отрицательных чисел. Если в нашей формуле встретиться у(0), то, поскольку его значение само по себе является неопределённым, мы должны подставить вместо у(0) рассматриваемое число n. Примеры:

    у(1) = у(0) =1 = 1
    у(2) = у(1) + у(0) = 1 + 2 = 3
    …

у(20) = у(19)+у(18)-у(15)-у(13)+9у(8)+у(5)=20+39-24-14+15+6= 42 Доказательство теоремы (**) я приводить не буду.

Вообще, найти сумму всех делителей числа можно с помощью канонического разложения натурального числа (это уже было сказано выше). Сумму делителей числа n обозначают у(n). Легко найти у(n) для небольших натуральных чисел, например у(12) = 1+2+3+4+6+12=28(это было приведено выше). Но при достаточно больших числах отыскивание всех делителей, а тем более их суммы становится затруднительным. Совсем другое дело, если уже известно, что каноническое разложение числа n таково: .

Его делителями являются все числа , для которых 0 ? вs ? бs, s = 1, …, k. Ясно, что у(n) представляет собой сумму всех таких чисел при различных значениях показателей

в1, в2, … вk. Этот результат мы получим раскрыв скобки в произведении

По формуле конечного числа членов геометрической прогрессии приходим к равенству

    (*)
    По этой формуле у(360) = .

Формулу для вычисления значения функции у(n) вывел замечательный английский математик Джон Валлис(1616 - 1703)–один из основателей и первых членов Лондонского Королевства общества (Академии наук). Он был первым из английских математиков, начавших заниматься математическим анализом. Ему принадлежат многие обозначения и термины, применяемые сейчас в математике, в частности знак? для обозначения бесконечности. Валлис вывел удивительную формулу, представляющую число р в виде бесконечного произведения:

Д. Валлис много занимался комбинаторикой и её приложениями к теории шифров, не без основания считая себя родоначальником новой науки–криптологии (от греч. “криптос” - тайный, “логос” - наука, учение). Он был одним из лучших шифровальщиков своего времени и по поручению министра полиции Терло занимался в республиканском правительстве Кромвеля расшифровкой посланий монархических заговорщиков.

    С функцией у(n) связан ряд любопытных задач. Например:

1. ) Найти пару целых чисел, удовлетворяющих условию: у(m1)=m2, у(m2)=m1. Некоторые из них не удаётся решить даже с использованием формулы (*). Так, например, не иначе как подбором можно найти числа, для которых у(n) есть квадрат некоторого натурального числа. Такими числами являются 22, 66, 70, 81, 343, 1501, 4479865. Вот ещё две задачи, приведённые в 1657 г. Пьером Ферма: найти такое m, для которого у(m3) – квадрат натурального числа (Ферма нашёл не одно решение этой задачи); найти такое m, для которого у(m2) – куб натурального числа. Например, одним из решений первой задачи является m = 7, а для второй m = 43098.

С помощью программы Derive, я попробовал найти ещё решения и у меня этого не получилось. (я рассматривал у(m3) = n2, где m принимает значения от 1 до 1000, а n от 1 до 5000 в 1. ) и тоже самое в 2. ) )

    Формулы:
    1. DELITELI(m) : = SELECT(MOD(m, n) = 0, n, 1, m)
    DIMENSION(DELITELI(m))
    2. SUMMADELITELEY(m) : = У ELEMENT(DELITELI(m), i)
    i=1



(C) 2009