Банк рефератов содержит более 364 тысяч рефератов, курсовых и дипломных работ, шпаргалок и докладов по различным дисциплинам: истории, психологии, экономике, менеджменту, философии, праву, экологии. А также изложения, сочинения по литературе, отчеты по практике, топики по английскому.
Полнотекстовый поиск
Всего работ:
364150
Теги названий
Разделы
Авиация и космонавтика (304)
Административное право (123)
Арбитражный процесс (23)
Архитектура (113)
Астрология (4)
Астрономия (4814)
Банковское дело (5227)
Безопасность жизнедеятельности (2616)
Биографии (3423)
Биология (4214)
Биология и химия (1518)
Биржевое дело (68)
Ботаника и сельское хоз-во (2836)
Бухгалтерский учет и аудит (8269)
Валютные отношения (50)
Ветеринария (50)
Военная кафедра (762)
ГДЗ (2)
География (5275)
Геодезия (30)
Геология (1222)
Геополитика (43)
Государство и право (20403)
Гражданское право и процесс (465)
Делопроизводство (19)
Деньги и кредит (108)
ЕГЭ (173)
Естествознание (96)
Журналистика (899)
ЗНО (54)
Зоология (34)
Издательское дело и полиграфия (476)
Инвестиции (106)
Иностранный язык (62792)
Информатика (3562)
Информатика, программирование (6444)
Исторические личности (2165)
История (21320)
История техники (766)
Кибернетика (64)
Коммуникации и связь (3145)
Компьютерные науки (60)
Косметология (17)
Краеведение и этнография (588)
Краткое содержание произведений (1000)
Криминалистика (106)
Криминология (48)
Криптология (3)
Кулинария (1167)
Культура и искусство (8485)
Культурология (537)
Литература : зарубежная (2044)
Литература и русский язык (11657)
Логика (532)
Логистика (21)
Маркетинг (7985)
Математика (3721)
Медицина, здоровье (10549)
Медицинские науки (88)
Международное публичное право (58)
Международное частное право (36)
Международные отношения (2257)
Менеджмент (12491)
Металлургия (91)
Москвоведение (797)
Музыка (1338)
Муниципальное право (24)
Налоги, налогообложение (214)
Наука и техника (1141)
Начертательная геометрия (3)
Оккультизм и уфология (8)
Остальные рефераты (21697)
Педагогика (7850)
Политология (3801)
Право (682)
Право, юриспруденция (2881)
Предпринимательство (475)
Прикладные науки (1)
Промышленность, производство (7100)
Психология (8694)
психология, педагогика (4121)
Радиоэлектроника (443)
Реклама (952)
Религия и мифология (2967)
Риторика (23)
Сексология (748)
Социология (4876)
Статистика (95)
Страхование (107)
Строительные науки (7)
Строительство (2004)
Схемотехника (15)
Таможенная система (663)
Теория государства и права (240)
Теория организации (39)
Теплотехника (25)
Технология (624)
Товароведение (16)
Транспорт (2652)
Трудовое право (136)
Туризм (90)
Уголовное право и процесс (406)
Управление (95)
Управленческие науки (24)
Физика (3463)
Физкультура и спорт (4482)
Философия (7216)
Финансовые науки (4592)
Финансы (5386)
Фотография (3)
Химия (2244)
Хозяйственное право (23)
Цифровые устройства (29)
Экологическое право (35)
Экология (4517)
Экономика (20645)
Экономико-математическое моделирование (666)
Экономическая география (119)
Экономическая теория (2573)
Этика (889)
Юриспруденция (288)
Языковедение (148)
Языкознание, филология (1140)

Реферат: Сумма делителей числа

Название: Сумма делителей числа
Раздел: Рефераты по математике
Тип: реферат Добавлен 00:19:06 22 августа 2005 Похожие работы
Просмотров: 789 Комментариев: 3 Оценило: 0 человек Средний балл: 0 Оценка: неизвестно     Скачать

.

Для начало приведём экспериментальный материал (который был получен с помощью программы Derive (по формуле 1.(см.ниже)): для нахождения делителей числа «a», программа делила число «a» на другие числа не превосходящие само число и если остаток от деления был равен 0, то число записывалось как делитель «a». ):

Ниже приведены все делители чисел от 1 до 1000:


[1, [1]]

[2, [1, 2]]

[3, [1, 3]]

[4, [1, 2, 4]]

[5, [1, 5]]

[6, [1, 2, 3, 6]]

[7, [1, 7]]

[8, [1, 2, 4, 8]]

[9, [1, 3, 9]]

[10, [1, 2, 5, 10]]

[11, [1, 11]]

[12, [1, 2, 3, 4, 6, 12]]

[13, [1, 13]]

[14, [1, 2, 7, 14]]

[15, [1, 3, 5, 15]]

[16, [1, 2, 4, 8, 16]]

[17, [1, 17]]

[18, [1, 2, 3, 6, 9, 18]]

[19, [1, 19]]

[20, [1, 2, 4, 5, 10, 20]]

[21, [1, 3, 7, 21]]

[22, [1, 2, 11, 22]]

[23, [1, 23]]

[24, [1, 2, 3, 4, 6, 8, 12, 24]]

[25, [1, 5, 25]]

[26, [1, 2, 13, 26]]

[27, [1, 3, 9, 27]]

[28, [1, 2, 4, 7, 14, 28]]

[29, [1, 29]]

[30, [1, 2, 3, 5, 6, 10, 15, 30]]

[31, [1, 31]]

[32, [1, 2, 4, 8, 16, 32]]

[33, [1, 3, 11, 33]]

[34, [1, 2, 17, 34]]

[35, [1, 5, 7, 35]]

[36, [1, 2, 3, 4, 6, 9, 12, 18, 36]]

[37, [1, 37]]

[38, [1, 2, 19, 38]]

[39, [1, 3, 13, 39]]

[40, [1, 2, 4, 5, 8, 10, 20, 40]]

[41, [1, 41]]

[42, [1, 2, 3, 6, 7, 14, 21, 42]]

[43, [1, 43]]

[44, [1, 2, 4, 11, 22, 44]]

[45, [1, 3, 5, 9, 15, 45]]

[46, [1, 2, 23, 46]]

[47, [1, 47]]

[48, [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]]

[49, [1, 7, 49]]

[50, [1, 2, 5, 10, 25, 50]]

[51, [1, 3, 17, 51]]

[52, [1, 2, 4, 13, 26, 52]]

[53, [1, 53]]

[54, [1, 2, 3, 6, 9, 18, 27, 54]]

[55, [1, 5, 11, 55]]

[56, [1, 2, 4, 7, 8, 14, 28, 56]]

[57, [1, 3, 19, 57]]

[58, [1, 2, 29, 58]]

[59, [1, 59]]

[60, [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]]

[61, [1, 61]]

[62, [1, 2, 31, 62]]

[63, [1, 3, 7, 9, 21, 63]]

[64, [1, 2, 4, 8, 16, 32, 64]]

[65, [1, 5, 13, 65]]

[66, [1, 2, 3, 6, 11, 22, 33, 66]]

[67, [1, 67]]

[68, [1, 2, 4, 17, 34, 68]]

[69, [1, 3, 23, 69]]

[70, [1, 2, 5, 7, 10, 14, 35, 70]]

[71, [1, 71]]

[72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]]

[73, [1, 73]]

[74, [1, 2, 37, 74]]

[75, [1, 3, 5, 15, 25, 75]]

[76, [1, 2, 4, 19, 38, 76]]

[77, [1, 7, 11, 77]]

[78, [1, 2, 3, 6, 13, 26, 39, 78]]

[79, [1, 79]]

[80, [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]]

[81, [1, 3, 9, 27, 81]]

[82, [1, 2, 41, 82]]

[83, [1, 83]]

[84, [1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84]]

[85, [1, 5, 17, 85]]

[86, [1, 2, 43, 86]]

[87, [1, 3, 29, 87]]

[88, [1, 2, 4, 8, 11, 22, 44, 88]]

[89, [1, 89]]

[90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90]]

[91, [1, 7, 13, 91]]

[92, [1, 2, 4, 23, 46, 92]]

[93, [1, 3, 31, 93]]

[94, [1, 2, 47, 94]]

[95, [1, 5, 19, 95]]

[96, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96]]

[97, [1, 97]]

[98, [1, 2, 7, 14, 49, 98]]

[99, [1, 3, 9, 11, 33, 99]]

[100, [1, 2, 4, 5, 10, 20, 25, 50, 100]]

[101, [1, 101]]

[102, [1, 2, 3, 6, 17, 34, 51, 102]]

[103, [1, 103]]

[104, [1, 2, 4, 8, 13, 26, 52, 104]]

[105, [1, 3, 5, 7, 15, 21, 35, 105]]

[106, [1, 2, 53, 106]]

[107, [1, 107]]

[108, [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108]]

[109, [1, 109]]

[110, [1, 2, 5, 10, 11, 22, 55, 110]]

[111, [1, 3, 37, 111]]

[112, [1, 2, 4, 7, 8, 14, 16, 28, 56, 112]]

[113, [1, 113]]

[114, [1, 2, 3, 6, 19, 38, 57, 114]]

[115, [1, 5, 23, 115]]

[116, [1, 2, 4, 29, 58, 116]]

[117, [1, 3, 9, 13, 39, 117]]

[118, [1, 2, 59, 118]]

[119, [1, 7, 17, 119]]

[120, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120]]

[121, [1, 11, 121]]

[122, [1, 2, 61, 122]]

[123, [1, 3, 41, 123]]

[124, [1, 2, 4, 31, 62, 124]]

[125, [1, 5, 25, 125]]

[126, [1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126]]

[127, [1, 127]]

[128, [1, 2, 4, 8, 16, 32, 64, 128]]

[129, [1, 3, 43, 129]]

[130, [1, 2, 5, 10, 13, 26, 65, 130]]

[131, [1, 131]]

[132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132]]

[133, [1, 7, 19, 133]]

[134, [1, 2, 67, 134]]

[135, [1, 3, 5, 9, 15, 27, 45, 135]]

[136, [1, 2, 4, 8, 17, 34, 68, 136]]

[137, [1, 137]]

[138, [1, 2, 3, 6, 23, 46, 69, 138]]

[139, [1, 139]]

[140, [1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140]]

[141, [1, 3, 47, 141]]

[142, [1, 2, 71, 142]]

[143, [1, 11, 13, 143]]

[144, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144]]

[145, [1, 5, 29, 145]]

[146, [1, 2, 73, 146]]

[147, [1, 3, 7, 21, 49, 147]]

[148, [1, 2, 4, 37, 74, 148]]

[149, [1, 149]]

[150, [1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150]]

[151, [1, 151]]

[152, [1, 2, 4, 8, 19, 38, 76, 152]]

[153, [1, 3, 9, 17, 51, 153]]

[154, [1, 2, 7, 11, 14, 22, 77, 154]]

[155, [1, 5, 31, 155]]

[156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156]]

[157, [1, 157]]

[158, [1, 2, 79, 158]]

[159, [1, 3, 53, 159]]

[160, [1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160]]

[161, [1, 7, 23, 161]]

[162, [1, 2, 3, 6, 9, 18, 27, 54, 81, 162]]

[163, [1, 163]]

[164, [1, 2, 4, 41, 82, 164]]

[165, [1, 3, 5, 11, 15, 33, 55, 165]]

[166, [1, 2, 83, 166]]

[167, [1, 167]]

[168, [1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168]]

[169, [1, 13, 169]]

[170, [1, 2, 5, 10, 17, 34, 85, 170]]

[171, [1, 3, 9, 19, 57, 171]]

[172, [1, 2, 4, 43, 86, 172]]

[173, [1, 173]]

[174, [1, 2, 3, 6, 29, 58, 87, 174]]

[175, [1, 5, 7, 25, 35, 175]]

[176, [1, 2, 4, 8, 11, 16, 22, 44, 88, 176]]

[177, [1, 3, 59, 177]]

[178, [1, 2, 89, 178]]

[179, [1, 179]]

[180, [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180]]

[181, [1, 181]]

[182, [1, 2, 7, 13, 14, 26, 91, 182]]

[183, [1, 3, 61, 183]]

[184, [1, 2, 4, 8, 23, 46, 92, 184]]

[185, [1, 5, 37, 185]]

[186, [1, 2, 3, 6, 31, 62, 93, 186]]

[187, [1, 11, 17, 187]]

[188, [1, 2, 4, 47, 94, 188]]

[189, [1, 3, 7, 9, 21, 27, 63, 189]]

[190, [1, 2, 5, 10, 19, 38, 95, 190]]

[191, [1, 191]]

[192, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192]]

[193, [1, 193]]

[194, [1, 2, 97, 194]]

[195, [1, 3, 5, 13, 15, 39, 65, 195]]

[196, [1, 2, 4, 7, 14, 28, 49, 98, 196]]

[197, [1, 197]]

[198, [1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198]]

[199, [1, 199]]

[200, [1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200]]

[201, [1, 3, 67, 201]]

[202, [1, 2, 101, 202]]

[203, [1, 7, 29, 203]]

[204, [1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204]]

[205, [1, 5, 41, 205]]

[206, [1, 2, 103, 206]]

[207, [1, 3, 9, 23, 69, 207]]

[208, [1, 2, 4, 8, 13, 16, 26, 52, 104, 208]]

[209, [1, 11, 19, 209]]

[210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210]]

[211, [1, 211]]

[212, [1, 2, 4, 53, 106, 212]]

[213, [1, 3, 71, 213]]

[214, [1, 2, 107, 214]]

[215, [1, 5, 43, 215]]

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

[217, [1, 7, 31, 217]]

[218, [1, 2, 109, 218]]

[219, [1, 3, 73, 219]]

[220, [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220]]

[221, [1, 13, 17, 221]]

[222, [1, 2, 3, 6, 37, 74, 111, 222]]

[223, [1, 223]]

[224, [1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112, 224]]

[225, [1, 3, 5, 9, 15, 25, 45, 75, 225]]

[226, [1, 2, 113, 226]]

[227, [1, 227]]

[228, [1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, 228]]

[229, [1, 229]]

[230, [1, 2, 5, 10, 23, 46, 115, 230]]

[231, [1, 3, 7, 11, 21, 33, 77, 231]]

[232, [1, 2, 4, 8, 29, 58, 116, 232]]

[233, [1, 233]]

[234, [1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234]]

[235, [1, 5, 47, 235]]

[236, [1, 2, 4, 59, 118, 236]]

[237, [1, 3, 79, 237]]

[238, [1, 2, 7, 14, 17, 34, 119, 238]]

[239, [1, 239]]

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

[241, [1, 241]]

[242, [1, 2, 11, 22, 121, 242]]

[243, [1, 3, 9, 27, 81, 243]]

[244, [1, 2, 4, 61, 122, 244]]

[245, [1, 5, 7, 35, 49, 245]]

[246, [1, 2, 3, 6, 41, 82, 123, 246]]

[247, [1, 13, 19, 247]]

[248, [1, 2, 4, 8, 31, 62, 124, 248]]

[249, [1, 3, 83, 249]]

[250, [1, 2, 5, 10, 25, 50, 125, 250]]

[251, [1, 251]]

[252, [1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252]]

[253, [1, 11, 23, 253]]

[254, [1, 2, 127, 254]]

[255, [1, 3, 5, 15, 17, 51, 85, 255]]

[256, [1, 2, 4, 8, 16, 32, 64, 128, 256]]

[257, [1, 257]]

[258, [1, 2, 3, 6, 43, 86, 129, 258]]

[259, [1, 7, 37, 259]]

[260, [1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260]]

[261, [1, 3, 9, 29, 87, 261]]

[262, [1, 2, 131, 262]]

[263, [1, 263]]

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

[265, [1, 5, 53, 265]]

[266, [1, 2, 7, 14, 19, 38, 133, 266]]

[267, [1, 3, 89, 267]]

[268, [1, 2, 4, 67, 134, 268]]

[269, [1, 269]]

[270, [1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270]]

[271, [1, 271]]

[272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272]]

[273, [1, 3, 7, 13, 21, 39, 91, 273]]

[274, [1, 2, 137, 274]]

[275, [1, 5, 11, 25, 55, 275]]

[276, [1, 2, 3, 4, 6, 12, 23, 46, 69, 92, 138, 276]]

[277, [1, 277]]

[278, [1, 2, 139, 278]]

[279, [1, 3, 9, 31, 93, 279]]

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

[281, [1, 281]]

[282, [1, 2, 3, 6, 47, 94, 141, 282]]

[283, [1, 283]]

[284, [1, 2, 4, 71, 142, 284]]

[285, [1, 3, 5, 15, 19, 57, 95, 285]]

[286, [1, 2, 11, 13, 22, 26, 143, 286]]

[287, [1, 7, 41, 287]]

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

[289, [1, 17, 289]]

[290, [1, 2, 5, 10, 29, 58, 145, 290]]

[291, [1, 3, 97, 291]]

[292, [1, 2, 4, 73, 146, 292]]

[293, [1, 293]]

[294, [1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294]]

[295, [1, 5, 59, 295]]

[296, [1, 2, 4, 8, 37, 74, 148, 296]]

[297, [1, 3, 9, 11, 27, 33, 99, 297]]

[298, [1, 2, 149, 298]]

[299, [1, 13, 23, 299]]

[300, [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300]]

[301, [1, 7, 43, 301]]

[302, [1, 2, 151, 302]]

[303, [1, 3, 101, 303]]

[304, [1, 2, 4, 8, 16, 19, 38, 76, 152, 304]]

[305, [1, 5, 61, 305]]

[306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306]]

[307, [1, 307]]

[308, [1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308]]

[309, [1, 3, 103, 309]]

[310, [1, 2, 5, 10, 31, 62, 155, 310]]

[311, [1, 311]]

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

[313, [1, 313]]

[314, [1, 2, 157, 314]]

[315, [1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315]]

[316, [1, 2, 4, 79, 158, 316]]

[317, [1, 317]]

[318, [1, 2, 3, 6, 53, 106, 159, 318]]

[319, [1, 11, 29, 319]]

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

[321, [1, 3, 107, 321]]

[322, [1, 2, 7, 14, 23, 46, 161, 322]]

[323, [1, 17, 19, 323]]

[324, [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324]]

[325, [1, 5, 13, 25, 65, 325]]

[326, [1, 2, 163, 326]]

[327, [1, 3, 109, 327]]

[328, [1, 2, 4, 8, 41, 82, 164, 328]]

[329, [1, 7, 47, 329]]

[330, [1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330]]

[331, [1, 331]]

[332, [1, 2, 4, 83, 166, 332]]

[333, [1, 3, 9, 37, 111, 333]]

[334, [1, 2, 167, 334]]

[335, [1, 5, 67, 335]]

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

[337, [1, 337]]

[338, [1, 2, 13, 26, 169, 338]]

[339, [1, 3, 113, 339]]

[340, [1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340]]

[341, [1, 11, 31, 341]]

[342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342]]

[343, [1, 7, 49, 343]]

[344, [1, 2, 4, 8, 43, 86, 172, 344]]

[345, [1, 3, 5, 15, 23, 69, 115, 345]]

[346, [1, 2, 173, 346]]

[347, [1, 347]]

[348, [1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, 348]]

[349, [1, 349]]

[350, [1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350]]

[351, [1, 3, 9, 13, 27, 39, 117, 351]]

[352, [1, 2, 4, 8, 11, 16, 22, 32, 44, 88, 176, 352]]

[353, [1, 353]]

[354, [1, 2, 3, 6, 59, 118, 177, 354]]

[355, [1, 5, 71, 355]]

[356, [1, 2, 4, 89, 178, 356]]

[357, [1, 3, 7, 17, 21, 51, 119, 357]]

[358, [1, 2, 179, 358]]

[359, [1, 359]]

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

[361, [1, 19, 361]]

[362, [1, 2, 181, 362]]

[363, [1, 3, 11, 33, 121, 363]]

[364, [1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182, 364]]

[365, [1, 5, 73, 365]]

[366, [1, 2, 3, 6, 61, 122, 183, 366]]

[367, [1, 367]]

[368, [1, 2, 4, 8, 16, 23, 46, 92, 184, 368]]

[369, [1, 3, 9, 41, 123, 369]]

[370, [1, 2, 5, 10, 37, 74, 185, 370]]

[371, [1, 7, 53, 371]]

[372, [1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, 372]]

[373, [1, 373]]

[374, [1, 2, 11, 17, 22, 34, 187, 374]]

[375, [1, 3, 5, 15, 25, 75, 125, 375]]

[376, [1, 2, 4, 8, 47, 94, 188, 376]]

[377, [1, 13, 29, 377]]

[378, [1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378]]

[379, [1, 379]]

[380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380]]

[381, [1, 3, 127, 381]]

[382, [1, 2, 191, 382]]

[383, [1, 383]]

[384, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384]]

[385, [1, 5, 7, 11, 35, 55, 77, 385]]

[386, [1, 2, 193, 386]]

[387, [1, 3, 9, 43, 129, 387]]

[388, [1, 2, 4, 97, 194, 388]]

[389, [1, 389]]

[390, [1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, 390]]

[391, [1, 17, 23, 391]]

[392, [1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, 392]]

[393, [1, 3, 131, 393]]

[394, [1, 2, 197, 394]]

[395, [1, 5, 79, 395]]

[396, [1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396]]

[397, [1, 397]]

[398, [1, 2, 199, 398]]

[399, [1, 3, 7, 19, 21, 57, 133, 399]]

[400, [1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400]]

[401, [1, 401]]

[402, [1, 2, 3, 6, 67, 134, 201, 402]]

[403, [1, 13, 31, 403]]

[404, [1, 2, 4, 101, 202, 404]]

[405, [1, 3, 5, 9, 15, 27, 45, 81, 135, 405]]

[406, [1, 2, 7, 14, 29, 58, 203, 406]]

[407, [1, 11, 37, 407]]

[408, [1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408]]

[409, [1, 409]]

[410, [1, 2, 5, 10, 41, 82, 205, 410]]

[411, [1, 3, 137, 411]]

[412, [1, 2, 4, 103, 206, 412]]

[413, [1, 7, 59, 413]]

[414, [1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414]]

[415, [1, 5, 83, 415]]

[416, [1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416]]

[417, [1, 3, 139, 417]]

[418, [1, 2, 11, 19, 22, 38, 209, 418]]

[419, [1, 419]]

[420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420]]

[421, [1, 421]]

[422, [1, 2, 211, 422]]

[423, [1, 3, 9, 47, 141, 423]]

[424, [1, 2, 4, 8, 53, 106, 212, 424]]

[425, [1, 5, 17, 25, 85, 425]]

[426, [1, 2, 3, 6, 71, 142, 213, 426]]

[427, [1, 7, 61, 427]]

[428, [1, 2, 4, 107, 214, 428]]

[429, [1, 3, 11, 13, 33, 39, 143, 429]]

[430, [1, 2, 5, 10, 43, 86, 215, 430]]

[431, [1, 431]]

[432, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432]]

[433, [1, 433]]

[434, [1, 2, 7, 14, 31, 62, 217, 434]]

[435, [1, 3, 5, 15, 29, 87, 145, 435]]

[436, [1, 2, 4, 109, 218, 436]]

[437, [1, 19, 23, 437]]

[438, [1, 2, 3, 6, 73, 146, 219, 438]]

[439, [1, 439]]

[440, [1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440]]

[441, [1, 3, 7, 9, 21, 49, 63, 147, 441]]

[442, [1, 2, 13, 17, 26, 34, 221, 442]]

[443, [1, 443]]

[444, [1, 2, 3, 4, 6, 12, 37, 74, 111, 148, 222, 444]]

[445, [1, 5, 89, 445]]

[446, [1, 2, 223, 446]]

[447, [1, 3, 149, 447]]

[448, [1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448]]

[449, [1, 449]]

[450, [1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450]]

[451, [1, 11, 41, 451]]

[452, [1, 2, 4, 113, 226, 452]]

[453, [1, 3, 151, 453]]

[454, [1, 2, 227, 454]]

[455, [1, 5, 7, 13, 35, 65, 91, 455]]

[456, [1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456]]

[457, [1, 457]]

[458, [1, 2, 229, 458]]

[459, [1, 3, 9, 17, 27, 51, 153, 459]]

[460, [1, 2, 4, 5, 10, 20, 23, 46, 92, 115, 230, 460]]

[461, [1, 461]]

[462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462]]

[463, [1, 463]]

[464, [1, 2, 4, 8, 16, 29, 58, 116, 232, 464]]

[465, [1, 3, 5, 15, 31, 93, 155, 465]]

[466, [1, 2, 233, 466]]

[467, [1, 467]]

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

[469, [1, 7, 67, 469]]

[470, [1, 2, 5, 10, 47, 94, 235, 470]]

[471, [1, 3, 157, 471]]

[472, [1, 2, 4, 8, 59, 118, 236, 472]]

[473, [1, 11, 43, 473]]

[474, [1, 2, 3, 6, 79, 158, 237, 474]]

[475, [1, 5, 19, 25, 95, 475]]

[476, [1, 2, 4, 7, 14, 17, 28, 34, 68, 119, 238, 476]]

[477, [1, 3, 9, 53, 159, 477]]

[478, [1, 2, 239, 478]]

[479, [1, 479]]

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

[481, [1, 13, 37, 481]]

[482, [1, 2, 241, 482]]

[483, [1, 3, 7, 21, 23, 69, 161, 483]]

[484, [1, 2, 4, 11, 22, 44, 121, 242, 484]]

[485, [1, 5, 97, 485]]

[486, [1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486]]

[487, [1, 487]]

[488, [1, 2, 4, 8, 61, 122, 244, 488]]

[489, [1, 3, 163, 489]]

[490, [1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]]

[491, [1, 491]]

[492, [1, 2, 3, 4, 6, 12, 41, 82, 123, 164, 246, 492]]

[493, [1, 17, 29, 493]]

[494, [1, 2, 13, 19, 26, 38, 247, 494]]

[495, [1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495]]

[496, [1, 2, 4, 8, 16, 31, 62, 124, 248, 496]]

[497, [1, 7, 71, 497]]

[498, [1, 2, 3, 6, 83, 166, 249, 498]]

[499, [1, 499]]

[500, [1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500]]

[501, [1, 3, 167, 501]]

[502, [1, 2, 251, 502]]

[503, [1, 503]]

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

[505, [1, 5, 101, 505]]

[506, [1, 2, 11, 22, 23, 46, 253, 506]]

[507, [1, 3, 13, 39, 169, 507]]

[508, [1, 2, 4, 127, 254, 508]]

[509, [1, 509]]

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

[511, [1, 7, 73, 511]]

[512, [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]]

[513, [1, 3, 9, 19, 27, 57, 171, 513]]

[514, [1, 2, 257, 514]]

[515, [1, 5, 103, 515]]

[516, [1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516]]

[517, [1, 11, 47, 517]]

[518, [1, 2, 7, 14, 37, 74, 259, 518]]

[519, [1, 3, 173, 519]]

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

[521, [1, 521]]

[522, [1, 2, 3, 6, 9, 18, 29, 58, 87, 174, 261, 522]]

[523, [1, 523]]

[524, [1, 2, 4, 131, 262, 524]]

[525, [1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525]]

[526, [1, 2, 263, 526]]

[527, [1, 17, 31, 527]]

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

[529, [1, 23, 529]]

[530, [1, 2, 5, 10, 53, 106, 265, 530]]

[531, [1, 3, 9, 59, 177, 531]]

[532, [1, 2, 4, 7, 14, 19, 28, 38, 76, 133, 266, 532]]

[533, [1, 13, 41, 533]]

[534, [1, 2, 3, 6, 89, 178, 267, 534]]

[535, [1, 5, 107, 535]]

[536, [1, 2, 4, 8, 67, 134, 268, 536]]

[537, [1, 3, 179, 537]]

[538, [1, 2, 269, 538]]

[539, [1, 7, 11, 49, 77, 539]]

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

[541, [1, 541]]

[542, [1, 2, 271, 542]]

[543, [1, 3, 181, 543]]

[544, [1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, 544]]

[545, [1, 5, 109, 545]]

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

[547, [1, 547]]

[548, [1, 2, 4, 137, 274, 548]]

[549, [1, 3, 9, 61, 183, 549]]

[550, [1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275, 550]]

[551, [1, 19, 29, 551]]

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

[553, [1, 7, 79, 553]]

[554, [1, 2, 277, 554]]

[555, [1, 3, 5, 15, 37, 111, 185, 555]]

[556, [1, 2, 4, 139, 278, 556]]

[557, [1, 557]]

[558, [1, 2, 3, 6, 9, 18, 31, 62, 93, 186, 279, 558]]

[559, [1, 13, 43, 559]]

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

[561, [1, 3, 11, 17, 33, 51, 187, 561]]

[562, [1, 2, 281, 562]]

[563, [1, 563]]

[564, [1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564]]

[565, [1, 5, 113, 565]]

[566, [1, 2, 283, 566]]

[567, [1, 3, 7, 9, 21, 27, 63, 81, 189, 567]]

[568, [1, 2, 4, 8, 71, 142, 284, 568]]

[569, [1, 569]]

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

[571, [1, 571]]

[572, [1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572]]

[573, [1, 3, 191, 573]]

[574, [1, 2, 7, 14, 41, 82, 287, 574]]

[575, [1, 5, 23, 25, 115, 575]]

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

[577, [1, 577]]

[578, [1, 2, 17, 34, 289, 578]]

[579, [1, 3, 193, 579]]

[580, [1, 2, 4, 5, 10, 20, 29, 58, 116, 145, 290, 580]]

[581, [1, 7, 83, 581]]

[582, [1, 2, 3, 6, 97, 194, 291, 582]]

[583, [1, 11, 53, 583]]

[584, [1, 2, 4, 8, 73, 146, 292, 584]]

[585, [1, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585]]

[586, [1, 2, 293, 586]]

[587, [1, 587]]

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

[589, [1, 19, 31, 589]]

[590, [1, 2, 5, 10, 59, 118, 295, 590]]

[591, [1, 3, 197, 591]]

[592, [1, 2, 4, 8, 16, 37, 74, 148, 296, 592]]

[593, [1, 593]]

[594, [1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594]]

[595, [1, 5, 7, 17, 35, 85, 119, 595]]

[596, [1, 2, 4, 149, 298, 596]]

[597, [1, 3, 199, 597]]

[598, [1, 2, 13, 23, 26, 46, 299, 598]]

[599, [1, 599]]

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

[601, [1, 601]]

[602, [1, 2, 7, 14, 43, 86, 301, 602]]

[603, [1, 3, 9, 67, 201, 603]]

[604, [1, 2, 4, 151, 302, 604]]

[605, [1, 5, 11, 55, 121, 605]]

[606, [1, 2, 3, 6, 101, 202, 303, 606]]

[607, [1, 607]]

[608, [1, 2, 4, 8, 16, 19, 32, 38, 76, 152, 304, 608]]

[609, [1, 3, 7, 21, 29, 87, 203, 609]]

[610, [1, 2, 5, 10, 61, 122, 305, 610]]

[611, [1, 13, 47, 611]]

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

[613, [1, 613]]

[614, [1, 2, 307, 614]]

[615, [1, 3, 5, 15, 41, 123, 205, 615]]

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

[617, [1, 617]]

[618, [1, 2, 3, 6, 103, 206, 309, 618]]

[619, [1, 619]]

[620, [1, 2, 4, 5, 10, 20, 31, 62, 124, 155, 310, 620]]

[621, [1, 3, 9, 23, 27, 69, 207, 621]]

[622, [1, 2, 311, 622]]

[623, [1, 7, 89, 623]]

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

[625, [1, 5, 25, 125, 625]]

[626, [1, 2, 313, 626]]

[627, [1, 3, 11, 19, 33, 57, 209, 627]]

[628, [1, 2, 4, 157, 314, 628]]

[629, [1, 17, 37, 629]]

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

[631, [1, 631]]

[632, [1, 2, 4, 8, 79, 158, 316, 632]]

[633, [1, 3, 211, 633]]

[634, [1, 2, 317, 634]]

[635, [1, 5, 127, 635]]

[636, [1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636]]

[637, [1, 7, 13, 49, 91, 637]]

[638, [1, 2, 11, 22, 29, 58, 319, 638]]

[639, [1, 3, 9, 71, 213, 639]]

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

[641, [1, 641]]

[642, [1, 2, 3, 6, 107, 214, 321, 642]]

[643, [1, 643]]

[644, [1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644]]

[645, [1, 3, 5, 15, 43, 129, 215, 645]]

[646, [1, 2, 17, 19, 34, 38, 323, 646]]

[647, [1, 647]]

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

[649, [1, 11, 59, 649]]

[650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650]]

[651, [1, 3, 7, 21, 31, 93, 217, 651]]

[652, [1, 2, 4, 163, 326, 652]]

[653, [1, 653]]

[654, [1, 2, 3, 6, 109, 218, 327, 654]]

[655, [1, 5, 131, 655]]

[656, [1, 2, 4, 8, 16, 41, 82, 164, 328, 656]]

[657, [1, 3, 9, 73, 219, 657]]

[658, [1, 2, 7, 14, 47, 94, 329, 658]]

[659, [1, 659]]

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

[661, [1, 661]]

[662, [1, 2, 331, 662]]

[663, [1, 3, 13, 17, 39, 51, 221, 663]]

[664, [1, 2, 4, 8, 83, 166, 332, 664]]

[665, [1, 5, 7, 19, 35, 95, 133, 665]]

[666, [1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666]]

[667, [1, 23, 29, 667]]

[668, [1, 2, 4, 167, 334, 668]]

[669, [1, 3, 223, 669]]

[670, [1, 2, 5, 10, 67, 134, 335, 670]]

[671, [1, 11, 61, 671]]

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

[673, [1, 673]]

[674, [1, 2, 337, 674]]

[675, [1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675]]

[676, [1, 2, 4, 13, 26, 52, 169, 338, 676]]

[677, [1, 677]]

[678, [1, 2, 3, 6, 113, 226, 339, 678]]

[679, [1, 7, 97, 679]]

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

[681, [1, 3, 227, 681]]

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

[683, [1, 683]]

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

[685, [1, 5, 137, 685]]

[686, [1, 2, 7, 14, 49, 98, 343, 686]]

[687, [1, 3, 229, 687]]

[688, [1, 2, 4, 8, 16, 43, 86, 172, 344, 688]]

[689, [1, 13, 53, 689]]

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

[691, [1, 691]]

[692, [1, 2, 4, 173, 346, 692]]

[693, [1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693]]

[694, [1, 2, 347, 694]]

[695, [1, 5, 139, 695]]

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

[697, [1, 17, 41, 697]]

[698, [1, 2, 349, 698]]

[699, [1, 3, 233, 699]]

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

[701, [1, 701]]

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

[703, [1, 19, 37, 703]]

[704, [1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704]]

[705, [1, 3, 5, 15, 47, 141, 235, 705]]

[706, [1, 2, 353, 706]]

[707, [1, 7, 101, 707]]

[708, [1, 2, 3, 4, 6, 12, 59, 118, 177, 236, 354, 708]]

[709, [1, 709]]

[710, [1, 2, 5, 10, 71, 142, 355, 710]]

[711, [1, 3, 9, 79, 237, 711]]

[712, [1, 2, 4, 8, 89, 178, 356, 712]]

[713, [1, 23, 31, 713]]

[714, [1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714]]

[715, [1, 5, 11, 13, 55, 65, 143, 715]]

[716, [1, 2, 4, 179, 358, 716]]

[717, [1, 3, 239, 717]]

[718, [1, 2, 359, 718]]

[719, [1, 719]]

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

[721, [1, 7, 103, 721]]

[722, [1, 2, 19, 38, 361, 722]]

[723, [1, 3, 241, 723]]

[724, [1, 2, 4, 181, 362, 724]]

[725, [1, 5, 25, 29, 145, 725]]

[726, [1, 2, 3, 6, 11, 22, 33, 66, 121, 242, 363, 726]]

[727, [1, 727]]

[728, [1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, 728]]

[729, [1, 3, 9, 27, 81, 243, 729]]

[730, [1, 2, 5, 10, 73, 146, 365, 730]]

[731, [1, 17, 43, 731]]

[732, [1, 2, 3, 4, 6, 12, 61, 122, 183, 244, 366, 732]]

[733, [1, 733]]

[734, [1, 2, 367, 734]]

[735, [1, 3, 5, 7, 15, 21, 35, 49, 105, 147, 245, 735]]

[736, [1, 2, 4, 8, 16, 23, 32, 46, 92, 184, 368, 736]]

[737, [1, 11, 67, 737]]

[738, [1, 2, 3, 6, 9, 18, 41, 82, 123, 246, 369, 738]]

[739, [1, 739]]

[740, [1, 2, 4, 5, 10, 20, 37, 74, 148, 185, 370, 740]]

[741, [1, 3, 13, 19, 39, 57, 247, 741]]

[742, [1, 2, 7, 14, 53, 106, 371, 742]]

[743, [1, 743]]

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

[745, [1, 5, 149, 745]]

[746, [1, 2, 373, 746]]

[747, [1, 3, 9, 83, 249, 747]]

[748, [1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748]]

[749, [1, 7, 107, 749]]

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

[751, [1, 751]]

[752, [1, 2, 4, 8, 16, 47, 94, 188, 376, 752]]

[753, [1, 3, 251, 753]]

[754, [1, 2, 13, 26, 29, 58, 377, 754]]

[755, [1, 5, 151, 755]]

[756, [1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756]]

[757, [1, 757]]

[758, [1, 2, 379, 758]]

[759, [1, 3, 11, 23, 33, 69, 253, 759]]

[760, [1, 2, 4, 5, 8, 10, 19, 20, 38, 40, 76, 95, 152, 190, 380, 760]]

[761, [1, 761]]

[762, [1, 2, 3, 6, 127, 254, 381, 762]]

[763, [1, 7, 109, 763]]

[764, [1, 2, 4, 191, 382, 764]]

[765, [1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765]]

[766, [1, 2, 383, 766]]

[767, [1, 13, 59, 767]]

[768, [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768]]

[769, [1, 769]]

[770, [1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770]]

[771, [1, 3, 257, 771]]

[772, [1, 2, 4, 193, 386, 772]]

[773, [1, 773]]

[774, [1, 2, 3, 6, 9, 18, 43, 86, 129, 258, 387, 774]]

[775, [1, 5, 25, 31, 155, 775]]

[776, [1, 2, 4, 8, 97, 194, 388, 776]]

[777, [1, 3, 7, 21, 37, 111, 259, 777]]

[778, [1, 2, 389, 778]]

[779, [1, 19, 41, 779]]

[780, [1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780]]

[781, [1, 11, 71, 781]]

[782, [1, 2, 17, 23, 34, 46, 391, 782]]

[783, [1, 3, 9, 27, 29, 87, 261, 783]]

[784, [1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784]]

[785, [1, 5, 157, 785]]

[786, [1, 2, 3, 6, 131, 262, 393, 786]]

[787, [1, 787]]

[788, [1, 2, 4, 197, 394, 788]]

[789, [1, 3, 263, 789]]

[790, [1, 2, 5, 10, 79, 158, 395, 790]]

[791, [1, 7, 113, 791]]

[792, [1, 2, 3, 4, 6, 8, 9, 11, 12, 18, 22, 24, 33, 36, 44, 66, 72, 88, 99, 132, 198, 264, 396, 792]]

[793, [1, 13, 61, 793]]

[794, [1, 2, 397, 794]]

[795, [1, 3, 5, 15, 53, 159, 265, 795]]

[796, [1, 2, 4, 199, 398, 796]]

[797, [1, 797]]

[798, [1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798]]

[799, [1, 17, 47, 799]]

[800, [1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800]]

[801, [1, 3, 9, 89, 267, 801]]

[802, [1, 2, 401, 802]]

[803, [1, 11, 73, 803]]

[804, [1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804]]

[805, [1, 5, 7, 23, 35, 115, 161, 805]]

[806, [1, 2, 13, 26, 31, 62, 403, 806]]

[807, [1, 3, 269, 807]]

[808, [1, 2, 4, 8, 101, 202, 404, 808]]

[809, [1, 809]]

[810, [1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810]]

[811, [1, 811]]

[812, [1, 2, 4, 7, 14, 28, 29, 58, 116, 203, 406, 812]]

[813, [1, 3, 271, 813]]

[814, [1, 2, 11, 22, 37, 74, 407, 814]]

[815, [1, 5, 163, 815]]

[816, [1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816]]

[817, [1, 19, 43, 817]]

[818, [1, 2, 409, 818]]

[819, [1, 3, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819]]

[820, [1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820]]

[821, [1, 821]]

[822, [1, 2, 3, 6, 137, 274, 411, 822]]

[823, [1, 823]]

[824, [1, 2, 4, 8, 103, 206, 412, 824]]

[825, [1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825]]

[826, [1, 2, 7, 14, 59, 118, 413, 826]]

[827, [1, 827]]

[828, [1, 2, 3, 4, 6, 9, 12, 18, 23, 36, 46, 69, 92, 138, 207, 276, 414, 828]]

[829, [1, 829]]

[830, [1, 2, 5, 10, 83, 166, 415, 830]]

[831, [1, 3, 277, 831]]

[832, [1, 2, 4, 8, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832]]

[833, [1, 7, 17, 49, 119, 833]]

[834, [1, 2, 3, 6, 139, 278, 417, 834]]

[835, [1, 5, 167, 835]]

[836, [1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836]]

[837, [1, 3, 9, 27, 31, 93, 279, 837]]

[838, [1, 2, 419, 838]]

[839, [1, 839]]

[840, [1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840]]

[841, [1, 29, 841]]

[842, [1, 2, 421, 842]]

[843, [1, 3, 281, 843]]

[844, [1, 2, 4, 211, 422, 844]]

[845, [1, 5, 13, 65, 169, 845]]

[846, [1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846]]

[847, [1, 7, 11, 77, 121, 847]]

[848, [1, 2, 4, 8, 16, 53, 106, 212, 424, 848]]

[849, [1, 3, 283, 849]]

[850, [1, 2, 5, 10, 17, 25, 34, 50, 85, 170, 425, 850]]

[851, [1, 23, 37, 851]]

[852, [1, 2, 3, 4, 6, 12, 71, 142, 213, 284, 426, 852]]

[853, [1, 853]]

[854, [1, 2, 7, 14, 61, 122, 427, 854]]

[855, [1, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855]]

[856, [1, 2, 4, 8, 107, 214, 428, 856]]

[857, [1, 857]]

[858, [1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858]]

[859, [1, 859]]

[860, [1, 2, 4, 5, 10, 20, 43, 86, 172, 215, 430, 860]]

[861, [1, 3, 7, 21, 41, 123, 287, 861]]

[862, [1, 2, 431, 862]]

[863, [1, 863]]

[864, [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864]]

[865, [1, 5, 173, 865]]

[866, [1, 2, 433, 866]]

[867, [1, 3, 17, 51, 289, 867]]

[868, [1, 2, 4, 7, 14, 28, 31, 62, 124, 217, 434, 868]]

[869, [1, 11, 79, 869]]

[870, [1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435, 870]]

[871, [1, 13, 67, 871]]

[872, [1, 2, 4, 8, 109, 218, 436, 872]]

[873, [1, 3, 9, 97, 291, 873]]

[874, [1, 2, 19, 23, 38, 46, 437, 874]]

[875, [1, 5, 7, 25, 35, 125, 175, 875]]

[876, [1, 2, 3, 4, 6, 12, 73, 146, 219, 292, 438, 876]]

[877, [1, 877]]

[878, [1, 2, 439, 878]]

[879, [1, 3, 293, 879]]

[880, [1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 40, 44, 55, 80, 88, 110, 176, 220, 440, 880]]

[881, [1, 881]]

[882, [1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 441, 882]]

[883, [1, 883]]

[884, [1, 2, 4, 13, 17, 26, 34, 52, 68, 221, 442, 884]]

[885, [1, 3, 5, 15, 59, 177, 295, 885]]

[886, [1, 2, 443, 886]]

[887, [1, 887]]

[888, [1, 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, 444, 888]]

[889, [1, 7, 127, 889]]

[890, [1, 2, 5, 10, 89, 178, 445, 890]]

[891, [1, 3, 9, 11, 27, 33, 81, 99, 297, 891]]

[892, [1, 2, 4, 223, 446, 892]]

[893, [1, 19, 47, 893]]

[894, [1, 2, 3, 6, 149, 298, 447, 894]]

[895, [1, 5, 179, 895]]

[896, [1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 448, 896]]

[897, [1, 3, 13, 23, 39, 69, 299, 897]]

[898, [1, 2, 449, 898]]

[899, [1, 29, 31, 899]]

[900, [1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900]]

[901, [1, 17, 53, 901]]

[902, [1, 2, 11, 22, 41, 82, 451, 902]]

[903, [1, 3, 7, 21, 43, 129, 301, 903]]

[904, [1, 2, 4, 8, 113, 226, 452, 904]]

[905, [1, 5, 181, 905]]

[906, [1, 2, 3, 6, 151, 302, 453, 906]]

[907, [1, 907]]

[908, [1, 2, 4, 227, 454, 908]]

[909, [1, 3, 9, 101, 303, 909]]

[910, [1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910]]

[911, [1, 911]]

[912, [1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912]]

[913, [1, 11, 83, 913]]

[914, [1, 2, 457, 914]]

[[915, [1, 3, 5, 15, 61, 183, 305, 915]]

[916, [1, 2, 4, 229, 458, 916]]

[917, [1, 7, 131, 917]]

[918, [1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, 918]]

[919, [1, 919]]

[920, [1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920]]

[921, [1, 3, 307, 921]]

[922, [1, 2, 461, 922]]

[923, [1, 13, 71, 923]]

[924, [1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924]]

[925, [1, 5, 25, 37, 185, 925]]

[926, [1, 2, 463, 926]]

[927, [1, 3, 9, 103, 309, 927]]

[928, [1, 2, 4, 8, 16, 29, 32, 58, 116, 232, 464, 928]]

[929, [1, 929]]

[930, [1, 2, 3, 5, 6, 10, 15, 30, 31, 62, 93, 155, 186, 310, 465, 930]]

[931, [1, 7, 19, 49, 133, 931]]

[932, [1, 2, 4, 233, 466, 932]]

[933, [1, 3, 311, 933]]

[934, [1, 2, 467, 934]]

[935, [1, 5, 11, 17, 55, 85, 187, 935]]

[936, [1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 36, 39, 52, 72, 78, 104, 117, 156, 234, 312, 468, 936]]

[937, [1, 937]]

[938, [1, 2, 7, 14, 67, 134, 469, 938]]

[939, [1, 3, 313, 939]]

[940, [1, 2, 4, 5, 10, 20, 47, 94, 188, 235, 470, 940]]

[941, [1, 941]]

[942, [1, 2, 3, 6, 157, 314, 471, 942]]

[943, [1, 23, 41, 943]]

[944, [1, 2, 4, 8, 16, 59, 118, 236, 472, 944]]

[945, [1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945]]

[946, [1, 2, 11, 22, 43, 86, 473, 946]]

[947, [1, 947]]

[948, [1, 2, 3, 4, 6, 12, 79, 158, 237, 316, 474, 948]]

[949, [1, 13, 73, 949]]

[950, [1, 2, 5, 10, 19, 25, 38, 50, 95, 190, 475, 950]]

[951, [1, 3, 317, 951]]

[952, [1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952]]

[953, [1, 953]]

[954, [1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954]]

[955, [1, 5, 191, 955]]

[956, [1, 2, 4, 239, 478, 956]]

[957, [1, 3, 11, 29, 33, 87, 319, 957]]

[958, [1, 2, 479, 958]]

[959, [1, 7, 137, 959]]

[960, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960]]

[961, [1, 31, 961]]

[962, [1, 2, 13, 26, 37, 74, 481, 962]]

[963, [1, 3, 9, 107, 321, 963]]

[964, [1, 2, 4, 241, 482, 964]]

[965, [1, 5, 193, 965]]

[966, [1, 2, 3, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966]]

[967, [1, 967]]

[968, [1, 2, 4, 8, 11, 22, 44, 88, 121, 242, 484, 968]]

[969, [1, 3, 17, 19, 51, 57, 323, 969]]

[970, [1, 2, 5, 10, 97, 194, 485, 970]]

[971, [1, 971]]

[972, [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972]]

[973, [1, 7, 139, 973]]

[974, [1, 2, 487, 974]]

[975, [1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975]]

[976, [1, 2, 4, 8, 16, 61, 122, 244, 488, 976]]

[977, [1, 977]]

[978, [1, 2, 3, 6, 163, 326, 489, 978]]

[979, [1, 11, 89, 979]]

[980, [1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980]]

[981, [1, 3, 9, 109, 327, 981]]

[982, [1, 2, 491, 982]]

[983, [1, 983]]

[984, [1, 2, 3, 4, 6, 8, 12, 24, 41, 82, 123, 164, 246, 328, 492, 984]]

[985, [1, 5, 197, 985]]

[986, [1, 2, 17, 29, 34, 58, 493, 986]]

[987, [1, 3, 7, 21, 47, 141, 329, 987]]

[988, [1, 2, 4, 13, 19, 26, 38, 52, 76, 247, 494, 988]]

[989, [1, 23, 43, 989]]

[990, [1, 2, 3, 5, 6, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990]]

[991, [1, 991]]

[992, [1, 2, 4, 8, 16, 31, 32, 62, 124, 248, 496, 992]]

[993, [1, 3, 331, 993]]

[994, [1, 2, 7, 14, 71, 142, 497, 994]]

[995, [1, 5, 199, 995]]

[996, [1, 2, 3, 4, 6, 12, 83, 166, 249, 332, 498, 996]]

[997, [1, 997]]

[998, [1, 2, 499, 998]]

[999, [1, 3, 9, 27, 37, 111, 333, 999]]

[1000, [1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000]]]


Теперь несложно посчитать и сумму делителей чисел от 1 до 1000(которые тоже были получены с помощью программы Derive (по формуле 2.), теперь делители «a» просто складывались):


[1, 1]

[2, 3]

[3, 4]

[4, 7]

[5, 6]

[6, 12]

[7, 8]

[8, 15]

[9, 13]

[10, 18]

[11, 12]

[12, 28]

[13, 14]

[14, 24]

[15, 24]

[16, 31]

[17, 18]

[18, 39]

[19, 20]

[20, 42]

[21, 32]

[22, 36]

[23, 24]

[24, 60]

[25, 31]

[26, 42]

[27, 40]

[28, 56]

[29, 30]

[30, 72]

[31, 32]

[32, 63]

[33, 48]

[34, 54]

[35, 48]

[36, 91]

[37, 38]

[38, 60]

[39, 56]

[40, 90]

[41, 42]

[42, 96]

[43, 44]

[44, 84]

[45, 78]

[46, 72]

[47, 48]

[48, 124]

[49, 57]

[50, 93]

[51, 72]

[52, 98]

[53, 54]

[54, 120]

[55, 72]

[56, 120]

[57, 80]

[58, 90]

[59, 60]

[60, 168]

[61, 62]

[62, 96]

[63, 104]

[64, 127]

[65, 84]

[66, 144]

[67, 68]

[68, 126]

[69, 96]

[70, 144]

[71, 72]

[72, 195]

[73, 74]

[74, 114]

[75, 124]

[76, 140]

[77, 96]

[78, 168]

[79, 80]

[80, 186]

[81, 121]

[82, 126]

[83, 84]

[84, 224]

[85, 108]

[86, 132]

[87, 120]

[88, 180]

[89, 90]

[90, 234]

[91, 112]

[92, 168]

[93, 128]

[94, 144]

[95, 120]

[96, 252]

[97, 98]

[98, 171]

[99, 156]

[100, 217]

[101, 102]

[102, 216]

[103, 104]

[104, 210]

[105, 192]

[106, 162]

[107, 108]

[108, 280]

[109, 110]

[110, 216]

[111, 152]

[112, 248]

[113, 114]

[114, 240]

[115, 144]

[116, 210]

[117, 182]

[118, 180]

[119, 144]

[120, 360]

[121, 133]

[122, 186]

[123, 168]

[124, 224]

[125, 156]

[126, 312]

[127, 128]

[128, 255]

[129, 176]

[130, 252]

[131, 132]

[132, 336]

[133, 160]

[134, 204]

[135, 240]

[136, 270]

[137, 138]

[138, 288]

[139, 140]

[140, 336]

[141, 192]

[142, 216]

[143, 168]

[144, 403]

[145, 180]

[146, 222]

[147, 228]

[148, 266]

[149, 150]

[150, 372]

[151, 152]

[152, 300]

[153, 234]

[154, 288]

[155, 192]

[156, 392]

[157, 158]

[158, 240]

[159, 216]

[160, 378]

[161, 192]

[162, 363]

[163, 164]

[164, 294]

[165, 288]

[166, 252]

[167, 168]

[168, 480]

[169, 183]

[170, 324]

[171, 260]

[172, 308]

[173, 174]

[174, 360]

[175, 248]

[176, 372]

[177, 240]

[178, 270]

[179, 180]

[180, 546]

[181, 182]

[182, 336]

[183, 248]

[184, 360]

[185, 228]

[186, 384]

[187, 216]

[188, 336]

[189, 320]

[190, 360]

[191, 192]

[192, 508]

[193, 194]

[194, 294]

[195, 336]

[196, 399]

[197, 198]

[198, 468]

[199, 200]

[200, 465]

[201, 272]

[202, 306]

[203, 240]

[204, 504]

[205, 252]

[206, 312]

[207, 312]

[208, 434]

[209, 240]

[210, 576]

[211, 212]

[212, 378]

[213, 288]

[214, 324]

[215, 264]

[216, 600]

[217, 256]

[218, 330]

[219, 296]

[220, 504]

[221, 252]

[222, 456]

[223, 224]

[224, 504]

[225, 403]

[226, 342]

[227, 228]

[228, 560]

[229, 230]

[230, 432]

[231, 384]

[232, 450]

[233, 234]

[234, 546]

[235, 288]

[236, 420]

[237, 320]

[238, 432]

[239, 240]

[240, 744]

[241, 242]

[242, 399]

[243, 364]

[244, 434]

[245, 342]

[246, 504]

[247, 280]

[248, 480]

[249, 336]

[250, 468]

[251, 252]

[252, 728]

[253, 288]

[254, 384]

[255, 432]

[256, 511]

[257, 258]

[258, 528]

[259, 304]

[260, 588]

[261, 390]

[262, 396]

[263, 264]

[264, 720]

[265, 324]

[266, 480]

[267, 360]

[268, 476]

[269, 270]

[270, 720]

[271, 272]

[272, 558]

[273, 448]

[274, 414]

[275, 372]

[276, 672]

[277, 278]

[278, 420]

[279, 416]

[280, 720]

[281, 282]

[282, 576]

[283, 284]

[284, 504]

[285, 480]

[286, 504]

[287, 336]

[288, 819]

[289, 307]

[290, 540]

[291, 392]

[292, 518]

[293, 294]

[294, 684]

[295, 360]

[296, 570]

[297, 480]

[298, 450]

[299, 336]

[300, 868]

[301, 352]

[302, 456]

[303, 408]

[304, 620]

[305, 372]

[306, 702]

[307, 308]

[308, 672]

[309, 416]

[310, 576]

[311, 312]

[312, 840]

[313, 314]

[314, 474]

[315, 624]

[316, 560]

[317, 318]

[318, 648]

[319, 360]

[320, 762]

[321, 432]

[322, 576]

[323, 360]

[324, 847]

[325, 434]

[326, 492]

[327, 440]

[328, 630]

[329, 384]

[330, 864]

[331, 332]

[332, 588]

[333, 494]

[334, 504]

[335, 408]

[336, 992]

[337, 338]

[338, 549]

[339, 456]

[340, 756]

[341, 384]

[342, 780]

[343, 400]

[344, 660]

[345, 576]

[346, 522]

[347, 348]

[348, 840]

[349, 350]

[350, 744]

[351, 560]

[352, 756]

[353, 354]

[354, 720]

[355, 432]

[356, 630]

[357, 576]

[358, 540]

[359, 360]

[360, 1170]

[361, 381]

[362, 546]

[363, 532]

[364, 784]

[365, 444]

[366, 744]

[367, 368]

[368, 744]

[369, 546]

[370, 684]

[371, 432]

[372, 896]

[373, 374]

[374, 648]

[375, 624]

[376, 720]

[377, 420]

[378, 960]

[379, 380]

[380, 840]

[381, 512]

[382, 576]

[383, 384]

[384, 1020]

[385, 576]

[386, 582]

[387, 572]

[388, 686]

[389, 390]

[390, 1008]

[391, 432]

[392, 855]

[393, 528]

[394, 594]

[395, 480]

[396, 1092]

[397, 398]

[398, 600]

[399, 640]

[400, 961]

[401, 402]

[402, 816]

[403, 448]

[404, 714]

[405, 726]

[406, 720]

[407, 456]

[408, 1080]

[409, 410]

[410, 756]

[411, 552]

[412, 728]

[413, 480]

[414, 936]

[415, 504]

[416, 882]

[417, 560]

[418, 720]

[419, 420]

[420, 1344]

[421, 422]

[422, 636]

[423, 624]

[424, 810]

[425, 558]

[426, 864]

[427, 496]

[428, 756]

[429, 672]

[430, 792]

[431, 432]

[432, 1240]

[433, 434]

[434, 768]

[435, 720]

[436, 770]

[437, 480]

[438, 888]

[439, 440]

[440, 1080]

[441, 741]

[442, 756]

[443, 444]

[444, 1064]

[445, 540]

[446, 672]

[447, 600]

[448, 1016]

[449, 450]

[450, 1209]

[451, 504]

[452, 798]

[453, 608]

[454, 684]

[455, 672]

[456, 1200]

[457, 458]

[458, 690]

[459, 720]

[460, 1008]

[461, 462]

[462, 1152]

[463, 464]

[464, 930]

[465, 768]

[466, 702]

[467, 468]

[468, 1274]

[469, 544]

[470, 864]

[471, 632]

[472, 900]

[473, 528]

[474, 960]

[475, 620]

[476, 1008]

[477, 702]

[478, 720]

[479, 480]

[480, 1512]

[481, 532]

[482, 726]

[483, 768]

[484, 931]

[485, 588]

[486, 1092]

[487, 488]

[488, 930]

[489, 656]

[490, 1026]

[491, 492]

[492, 1176]

[493, 540]

[494, 840]

[495, 936]

[496, 992]

[497, 576]

[498, 1008]

[499, 500]

[500, 1092]

[501, 672]

[502, 756]

[503, 504]

[504, 1560]

[505, 612]

[506, 864]

[507, 732]

[508, 896]

[509, 510]

[510, 1296]

[511, 592]

[512, 1023]

[513, 800]

[514, 774]

[515, 624]

[516, 1232]

[517, 576]

[518, 912]

[519, 696]

[520, 1260]

[521, 522]

[522, 1170]

[523, 524]

[524, 924]

[525, 992]

[526, 792]

[527, 576]

[528, 1488]

[529, 553]

[530, 972]

[531, 780]

[532, 1120]

[533, 588]

[534, 1080]

[535, 648]

[536, 1020]

[537, 720]

[538, 810]

[539, 684]

[540, 1680]

[541, 542]

[542, 816]

[543, 728]

[544, 1134]

[545, 660]

[546, 1344]

[547, 548]

[548, 966]

[549, 806]

[550, 1116]

[551, 600]

[552, 1440]

[553, 640]

[554, 834]

[555, 912]

[556, 980]

[557, 558]

[558, 1248]

[559, 616]

[560, 1488]

[561, 864]

[562, 846]

[563, 564]

[564, 1344]

[565, 684]

[566, 852]

[567, 968]

[568, 1080]

[569, 570]

[570, 1440]

[571, 572]

[572, 1176]

[573, 768]

[574, 1008]

[575, 744]

[576, 1651]

[577, 578]

[578, 921]

[579, 776]

[580, 1260]

[581, 672]

[582, 1176]

[583, 648]

[584, 1110]

[585, 1092]

[586, 882]

[587, 588]

[588, 1596]

[589, 640]

[590, 1080]

[591, 792]

[592, 1178]

[593, 594]

[594, 1440]

[595, 864]

[596, 1050]

[597, 800]

[598, 1008]

[599, 600]

[600, 1860]

[601, 602]

[602, 1056]

[603, 884]

[604, 1064]

[605, 798]

[606, 1224]

[607, 608]

[608, 1260]

[609, 960]

[610, 1116]

[611, 672]

[612, 1638]

[613, 614]

[614, 924]

[615, 1008]

[616, 1440]

[617, 618]

[618, 1248]

[619, 620]

[620, 1344]

[621, 960]

[622, 936]

[623, 720]

[624, 1736]

[625, 781]

[626, 942]

[627, 960]

[628, 1106]

[629, 684]

[630, 1872]

[631, 632]

[632, 1200]

[633, 848]

[634, 954]

[635, 768]

[636, 1512]

[637, 798]

[638, 1080]

[639, 936]

[640, 1530]

[641, 642]

[642, 1296]

[643, 644]

[644, 1344]

[645, 1056]

[646, 1080]

[647, 648]

[648, 1815]

[649, 720]

[650, 1302]

[651, 1024]

[652, 1148]

[653, 654]

[654, 1320]

[655, 792]

[656, 1302]

[657, 962]

[658, 1152]

[659, 660]

[660, 2016]

[661, 662]

[662, 996]

[663, 1008]

[664, 1260]

[665, 960]

[666, 1482]

[667, 720]

[668, 1176]

[669, 896]

[670, 1224]

[671, 744]

[672, 2016]

[673, 674]

[674, 1014]

[675, 1240]

[676, 1281]

[677, 678]

[678, 1368]

[679, 784]

[680, 1620]

[681, 912]

[682, 1152]

[683, 684]

[684, 1820]

[685, 828]

[686, 1200]

[687, 920]

[688, 1364]

[689, 756]

[690, 1728]

[691, 692]

[692, 1218]

[693, 1248]

[694, 1044]

[695, 840]

[696, 1800]

[697, 756]

[698, 1050]

[699, 936]

[700, 1736]

[701, 702]

[702, 1680]

[703, 760]

[704, 1524]

[705, 1152]

[706, 1062]

[707, 816]

[708, 1680]

[709, 710]

[710, 1296]

[711, 1040]

[712, 1350]

[713, 768]

[714, 1728]

[715, 1008]

[716, 1260]

[717, 960]

[718, 1080]

[719, 720]

[720, 2418]

[721, 832]

[722, 1143]

[723, 968]

[724, 1274]

[725, 930]

[726, 1596]

[727, 728]

[728, 1680]

[729, 1093]

[730, 1332]

[731, 792]

[732, 1736]

[733, 734]

[734, 1104]

[735, 1368]

[736, 1512]

[737, 816]

[738, 1638]

[739, 740]

[740, 1596]

[741, 1120]

[742, 1296]

[743, 744]

[744, 1920]

[745, 900]

[746, 1122]

[747, 1092]

[748, 1512]

[749, 864]

[750, 1872]

[751, 752]

[752, 1488]

[753, 1008]

[754, 1260]

[755, 912]

[756, 2240]

[757, 758]

[758, 1140]

[759, 1152]

[760, 1800]

[761, 762]

[762, 1536]

[763, 880]

[764, 1344]

[765, 1404]

[766, 1152]

[767, 840]

[768, 2044]

[769, 770]

[770, 1728]

[771, 1032]

[772, 1358]

[773, 774]

[774, 1716]

[775, 992]

[776, 1470]

[777, 1216]

[778, 1170]

[779, 840]

[780, 2352]

[781, 864]

[782, 1296]

[783, 1200]

[784, 1767]

[785, 948]

[786, 1584]

[787, 788]

[788, 1386]

[789, 1056]

[790, 1440]

[791, 912]

[792, 2340]

[793, 868]

[794, 1194]

[795, 1296]

[796, 1400]

[797, 798]

[798, 1920]

[799, 864]

[800, 1953]

[801, 1170]

[802, 1206]

[803, 888]

[804, 1904]


[805, 1152]

[806, 1344]

[807, 1080]

[808, 1530]

[809, 810]

[810, 2178]

[811, 812]

[812, 1680]

[813, 1088]

[814, 1368]

[815, 984]

[816, 2232]

[817, 880]

[818, 1230]

[819, 1456]

[820, 1764]

[821, 822]

[822, 1656]

[823, 824]

[824, 1560]

[825, 1488]

[826, 1440]

[827, 828]

[828, 2184]

[829, 830]

[830, 1512]

[831, 1112]

[832, 1778]

[833, 1026]

[834, 1680]

[835, 1008]

[836, 1680]

[837, 1280]

[838, 1260]

[839, 840]

[840, 2880]

[841, 871]

[842, 1266]

[843, 1128]

[844, 1484]

[845, 1098]

[846, 1872]

[847, 1064]

[848, 1674]

[849, 1136]

[850, 1674]

[851, 912]

[852, 2016]

[853, 854]

[854, 1488]

[855, 1560]

[856, 1620]

[857, 858]

[858, 2016]

[859, 860]

[860, 1848]

[861, 1344]

[862, 1296]

[863, 864]

[864, 2520]

[865, 1044]

[866, 1302]

[867, 1228]

[868, 1792]

[869, 960]

[870, 2160]

[871, 952]

[872, 1650]

[873, 1274]

[874, 1440]

[875, 1248]

[876, 2072]

[877, 878]

[878, 1320]

[879, 1176]

[880, 2232]

[881, 882]

[882, 2223]

[883, 884]

[884, 1764]

[885, 1440]

[886, 1332]

[887, 888]

[888, 2280]

[889, 1024]

[890, 1620]

[891, 1452]

[892, 1568]

[893, 960]

[894, 1800]

[895, 1080]

[896, 2040]

[897, 1344]

[898, 1350]

[899, 960]

[900, 2821]

[901, 972]

[902, 1512]

[903, 1408]

[904, 1710]

[905, 1092]

[906, 1824]

[907, 908]

[908, 1596]

[909, 1326]

[910, 2016]

[911, 912]

[912, 2480]

[913, 1008]

[914, 1374]

[915, 1488]

[916, 1610]

[917, 1056]

[918, 2160]

[919, 920]

[920, 2160]

[921, 1232]

[922, 1386]

[923, 1008]

[924, 2688]

[925, 1178]

[926, 1392]

[927, 1352]

[928, 1890]

[929, 930]

[930, 2304]

[931, 1140]

[932, 1638]

[933, 1248]

[934, 1404]

[935, 1296]

[936, 2730]

[937, 938]

[938, 1632]

[939, 1256]

[940, 2016]

[941, 942]

[942, 1896]

[943, 1008]

[944, 1860]

[945, 1920]

[946, 1584]

[947, 948]

[948, 2240]

[949, 1036]

[950, 1860]

[951, 1272]

[952, 2160]

[953, 954]

[954, 2106]

[955, 1152]

[956, 1680]

[957, 1440]

[958, 1440]

[959, 1104]

[960, 3048]

[961, 993]

[962, 1596]

[963, 1404]

[964, 1694]

[965, 1164]

[966, 2304]

[967, 968]

[968, 1995]

[969, 1440]

[970, 1764]

[971, 972]

[972, 2548]

[973, 1120]

[974, 1464]

[975, 1736]

[976, 1922]

[977, 978]

[978, 1968]

[979, 1080]

[980, 2394]

[981, 1430]

[982, 1476]

[983, 984]

[984, 2520]

[985, 1188]

[986, 1620]

[987, 1536]

[988, 1960]

[989, 1056]

[990, 2808]

[991, 992]

[992, 2016]

[993, 1328]

[994, 1728]

[995, 1200]

[996, 2352]

[997, 998]

[998, 1500]

[999, 1520]

[1000, 2340]


Теперь посмотрим, все ли числа являются суммой делителей какого-либо числа и есть ли такие числа сумма делителей которых равна (в первых двух сотнях).

Ниже приведена таблица: [[4, 7]](на втором месте сумма делителей, а на первом число с данной суммой делителей) … [[1, 1]], [2] (т.е. нет такого числа с суммой делителей равной двум):


[1,1]

[2]

[2,3]

[3,4]

[5]

[5,6]

[4,7]

[7,8]

[9]

[10]

[11]

[6,12]

[11, 12]

[9,13]

[13,14]

[8,15]

[16]

[17]

[10,18]

[17,18]

[19]

[19.20]

[21]

[22]

[23]

[14,24]

[15,24]

[23,24]

[25]

[26]

[27]

[12, 28].

[29]

[29,30]

[16,31]

[25.31]

[21,32]

[31,32]

[33]

[34]

[35]

[22,36]

[37]

[37,38]

[18,39]

[27, 40]

[41]

[20,42]

[26,42]

[41,42].

[43]

[43,44].

[45]

[46]

[47]

[33,48].

[35,4 8]

[47,48]

[49]

[50]

[51]

[52]

[53]

[34,54]

[53, 54]

[55]

[28,56]

[39.56]

[49,57]

[58]

[59]

[24,60]

[38.60]

[59,60]

[61]

[61,62]

[32,63]

[64]

[65]

[66]

[67]

[67, 68]

[69]

[70]

[71]

[30,72]

[46,72]

[51,72]

[55,72]

[71,72]

[73]

[73,74]

[75]

[76]

[77]

[45,78]

[79]

[57,80]

[79,80]

[81]

[82]

[83]

[44,84]

[65,84]

[83,84]

[85]

[86]

[87]

[88]

[89]

[40, 90]

[58,90]

[89,90]

[36,91]

[92]

[50,93].

[94]

[95]

[42, 96]

[62,96]

[69,96]

[77,96]

[97]

[52,98]

[97,98]

[99]

[100]

[101]

[102]

[103]

[63,104]

[105]

[106]

[107]

[85,108]

[109]

[110]

[111]

[91, 112]

[113]

[74,114],

[115]

[116]

[117]

[118]

[119]

[54,120]

[56,120]

[87,120]

[95,120]

[81,121]

[122]

[123]

[48,124]

[75, 124]

[125]

[68,126]

[82.126]

[64,127]

[9 3,128]

[129]

[130]

[131]

[86,132]

[133]

[134]

[135]

[136]

[137]

[138]

[139]

[76,140]

[141]

[142]

[143]

[66,144]

[70,144]

[94,144]

[145]

[146]

[147]

[178]

[149]

[150]

[151]

[152]

[153]

[154]

[155]

[99,156]

[157]

[158]

[159]

[160]

[161]

[162]

[163]

[164]

[165]

[166]

[167]

[60,168]

[78,168]

[92,168]

[169]

[170]

[98,171]

[172]

[173]

[174]

[175]

[176]

[177]

[178]

[179]

[88,180]

[181]

[182]

[183]

[184]

[185]

[80,186]

[187]

[188]

[189]

[190]

[191]

[192]

[193]

[194]

[72,195]

[196]

[197]

[198]

[199]

[200]


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


[6,12], [11,12]

[10,18], [17,18]

[14,24], [15,24], [23,24]

[16,31]. [25,31]

[21,32], [31,32]

[20, 42], [26,42], [41,42]

[33,48], [35,48], [47,48]

[34,5 4], [53,54]

[28,56], [39,56]

[24,60], [38,60], [59, 60]

[30,72], [46,72], [51,72], [55,72], [71,72]

[57,80], [79,80]

[44,84], [65,84], [83,84]

[40,90], [58, 9 0], [89,90]

[42,96], [62,96], [69,96], [77,96]

[52,98], [97,98]

[54,120], [56, 120], [87,120], [95,120]

[48,124], [75,124]

[68,126], [82,126]

[66,144], [70, 144], [94,144]

[60,168], [78,168], [92,168]


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


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

Посмотрим, что же у нас получилось: на графике отчётливо просматриваются несколько прямых линий, например, нижняя это – простые числа. Верхняя граница – это наиболее сложные числа (имеющие наибольшее количество делителей) - это не прямая, но и не парабола. Скорее всего, – это показательная функция (у = ах ).

В мемуарах Эйлера я нашел много интересных мне рассуждений(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(aq bw ce dr )= 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:

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

Оценить/Добавить комментарий
Имя
Оценка
Комментарии:
Где скачать еще рефератов? Здесь: letsdoit777.blogspot.com
Евгений22:24:22 18 марта 2016
Кто еще хочет зарабатывать от 9000 рублей в день "Чистых Денег"? Узнайте как: business1777.blogspot.com ! Cпециально для студентов!
11:54:31 24 ноября 2015
класс
даша19:07:22 11 октября 2009

Работы, похожие на Реферат: Сумма делителей числа

Назад
Меню
Главная
Рефераты
Благодарности
Опрос
Станете ли вы заказывать работу за деньги, если не найдете ее в Интернете?

Да, в любом случае.
Да, но только в случае крайней необходимости.
Возможно, в зависимости от цены.
Нет, напишу его сам.
Нет, забью.



Результаты(151191)
Комментарии (1843)
Copyright © 2005-2016 BestReferat.ru bestreferat@mail.ru       реклама на сайте

Рейтинг@Mail.ru