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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]]