Сумма делителей числа

стр. 1 из 3

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.

     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(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:

     

	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 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:

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

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