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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[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(?(n) –
сумма делителей числа n): Определив значение ?(n) мы ясно видим, что если p
– простое, то ?(p)= p + 1. ?(1)=1, а если число n – составное, то ?(n)>1 +
n.
Если a, b, c, d – различные простые числа, то мы видим:
?(ab)=1+a+b+ab=(1+a)(1+b)= ?(a)?(b)
?(abcd)= ?(a)?(b)?(c)?(d)
?(a^2)=1+a+a2=[pic]
?(a^3)=1+a+a2+a3=[pic]
И вообще
?(nn)=[pic]
Пользуясь этим:
?(aqbwcedr)= ?(aq)?(bw)?(ce)?(dr)
Например ?(360), 360 = 23*32*5 => ?(23) ?(32) ?(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 ?(n) обозначает член любой этой последовательности, а ?(n - 1),
?(n - 2), ?(n - 3)… предшествующие члены, то ?(n) всегда можно получить по
нескольким предыдущим членам:
?(n) = ?(n - 1) + ?(n - 2) - ?(n - 5) - ?(n - 7) + ?(n - 12) + ?(n -
15) - ?(n - 22) - ?(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 ?, еще
положительны, и опускать ? для отрицательных чисел. Если в нашей формуле
встретиться ?(0), то, поскольку его значение само по себе является
неопределённым, мы должны подставить вместо ?(0) рассматриваемое число n.
Примеры:
?(1) = ?(0) =1
= 1
?(2) = ?(1) + ?(0) = 1 + 2
= 3
…
?(20) = ?(19)+?(18)-?(15)-?(13)+9?(8)+?(5)=20+39-24-14+15+6= 42
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 ?(n). Легко найти ?(n) для небольших натуральных чисел,
например ?(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:[pic].
Его делителями являются все числа [pic], для которых 0 ? ?s ? ?s, s = 1, …,
k. Ясно, что ?(n) представляет собой сумму всех таких чисел при различных
значениях показателей
?1, ?2, … ?k. Этот результат мы получим раскрыв скобки в произведении
[pic]
По формуле конечного числа членов геометрической прогрессии приходим к
равенству
[pic] (*)
По этой формуле ?(360) = [pic] .
Формулу для вычисления значения функции ?(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 ? в виде бесконечного произведения:
[pic]
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 ?(n) связан ряд любопытных задач. Например:
1.) Найти пару целых чисел, удовлетворяющих условию: ?(m1)=m2,
?(m2)=m1.
Некоторые из них не удаётся решить даже с использованием формулы (*).
Так, например, не иначе как подбором можно найти числа, для которых ?(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 ?(m3) – квадрат натурального числа (Ферма нашёл не одно решение этой задачи);
2. найти такое m, для которого ?(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 ?(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) := ?
ELEMENT(DELITELI(m), i) i=1
(zip - application/zip)
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