Реферат: Сумма делителей числа

Реферат: 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]]

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[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.

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(σ(n) – сумма делителей числа n): Определив значение σ(n) мы ясно видим, что если p – простое, то σ(p)= p + 1. σ(1)=1, а если число n – составное, то σ(n)>1 + n.

     Если a, b, c, d – различные простые числа, то мы видим:

     σ(ab)=1+a+b+ab=(1+a)(1+b)= σ(a)σ(b)

     σ(abcd)= σ(a)σ(b)σ(c)σ(d)

     σ(a^2)=1+a+a2=

     σ(a^3)=1+a+a2+a3=

     И вообще

     σ(nn)=

     Пользуясь этим:

     σ(aqbwcedr)= σ(aq)σ(bw)σ(ce)σ(dr)

     Например σ(360), 360 = 23*32*5 => σ(23) σ(32) σ(5)=15*13*6=1170.

     Чтобы показать последовательность сумм делителей приведём таблицу:

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

     σ(n) = σ(n - 1) + σ(n - 2) - σ(n - 5) - σ(n - 7) + σ(n - 12) + σ(n - 15) - σ(n - 22) - σ(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 σ, еще положительны, и опускать σ для отрицательных чисел. Если в нашей формуле встретиться σ(0), то, поскольку его значение само по себе является неопределённым, мы должны подставить вместо σ(0) рассматриваемое число n. Примеры:

 σ(1)  = σ(0)                                                     =1                              = 1 

 σ(2)  = σ(1) + σ(0)                                          = 1 + 2                       = 3

…     

σ(20) = σ(19)+σ(18)-σ(15)-σ(13)+9σ(8)+σ(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 σ(n). Легко найти σ(n) для небольших натуральных чисел, например σ(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 ≤ βs ≤ αs, s = 1, …, k. Ясно, что σ(n) представляет собой сумму всех таких чисел при различных значениях показателей

β1, β2, … βk. Этот результат мы получим раскрыв скобки в произведении

По формуле конечного числа членов геометрической прогрессии приходим к равенству

                                                                                       (*)

По этой формуле σ(360) =  .                                                      

Формулу для вычисления значения функции σ(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 π 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 σ(n) связан ряд любопытных задач. Например:

    1.) Найти пару целых чисел, удовлетворяющих условию: σ(m1)=m2, σ(m2)=m1.

      Некоторые из них не удаётся решить даже с использованием формулы (*). Так, например, не иначе как подбором можно найти числа, для которых σ(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:

1.)   найти такое m, для которого σ(m3) – квадрат натурального числа (Ферма нашёл не одно решение этой задачи);

2.)   найти такое m, для которого σ(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 σ(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) :=                     Σ                         ELEMENT(DELITELI(m), i)

                                                                       i=1