Как найти сумму делителей нескольких чисел

Онлайн калькулятор поможет вычислить сумму простых делителей числа, выпишет все делители, число положительных делителей натурального числа.

Сумма простых делителей числа
Число

Given a natural number n, the task is to find sum of divisors of all the divisors of n.

Examples: 

Input : n = 54
Output : 232
Divisors of 54 = 1, 2, 3, 6, 9, 18, 27, 54.
Sum of divisors of 1, 2, 3, 6, 9, 18, 27, 54 
are 1, 3, 4, 12, 13, 39, 40, 120 respectively.
Sum of divisors of all the divisors of 54 = 
1 + 3 + 4 + 12 + 13 + 39 + 40 + 120 = 232.

Input : n = 10
Output : 28
Divisors of 10 are 1, 2, 5, 10
Sums of divisors of divisors are 
1, 3, 6, 18.
Overall sum = 1 + 3 + 6 + 18 = 28

Using the fact that any number n can be expressed as product of prime factors, n = p₁^k1 x p₂^k2 x … where p₁, p₂, … are prime numbers. 
All the divisors of n can be expressed as p₁^a x p₂^b x …, where 0 <= a <= k1 and 0 <= b <= k2. 
Now sum of divisors will be sum of all power of p₁ – p₁⁰, p₁¹,…., p₁^k1 multiplied by all power of p₂ – p₂⁰, p₂¹,…., p₂^k1 
Sum of Divisor of n 
= (p₁⁰ x p₂⁰) + (p₁¹ x p₂⁰) +…..+ (p₁^k1 x p₂⁰) +….+ (p₁⁰ x p₂¹) + (p₁¹ x p₂¹) +…..+ (p₁^k1 x p₂¹) +……..+ 
   (p₁⁰ x p₂^k2) + (p₁¹ x p₂^k2) +……+ (p₁^k1 x p₂^k2). 
= (p₁⁰ + p₁¹ +…+ p₁^k1) x p₂⁰ + (p₁⁰ + p₁¹ +…+ p₁^k1) x p₂¹ +…….+ (p₁⁰ + p₁¹ +…+ p₁^k1) x p₂^k2. 
= (p₁⁰ + p₁¹ +…+ p₁^k1) x (p₂⁰ + p₂¹ +…+ p₂^k2).

Now, the divisors of any p^a, for p as prime, are p⁰, p¹,……, p^a. And sum of divisors will be (p^(a+1) – 1)/(p -1), let it define by f(p). 
So, sum of divisors of all divisor will be, 
= (f(p₁⁰) + f(p₁¹) +…+ f(p₁^k1)) x (f(p₂⁰) + f(p₂¹) +…+ f(p₂^k2)).

So, given a number n, by prime factorization we can find the sum of divisors of all the divisors. But in this problem we are given that n is product of element of array. So, find prime factorization of each element and by using the fact a^b x a^c = a^b+c.

Below is the implementation of this approach:  

Output: 

28

Time Complexity: O(√n log n) 
Auxiliary Space: O(n)

Optimizations : 
For the cases when there are multiple inputs for which we need find the value, we can use Sieve of Eratosthenes as discussed in this post.

This article is contributed by Aarti_Rathi and Anuj Chauhan. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

Last Updated :
23 Jun, 2022

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Число и сумма натуральных делителей натурального числа
Основная теорема арифметики. Всякое натуральное число п > 1 либо просто, либо может быть представлено, и притом единственным образом – с точностью до порядка следования сомножителей, в виде произведения простых чисел (можно считать, что любое натуральное число, большее 1, можно представить в виде произведения простых чисел, если считать , что это произведение может содержать всего лишь один множитель).
Среди простых сомножителей, присутствующих в разложении `n = p1*p2*…*pk`, могут быть и одинаковые. Например, `24=2*2*2*3`. Их можно объединить, воспользовавшись операцией возведения в степень. Кроме того, простые сомножители можно упорядочить по величине. В результате получается разложение
`n = p_1^(alpha_1)*p_2^(alpha_2)*…….*p_k^(alpha_k)`, где `alpha_1, alpha_2, ……, alpha_k in NN`
(1)
Такое представление числа называется каноническим разложением его на простые сомножители. Например, каноническое представление числа 2 520 имеет вид 2 520 = 2³ • З² • 5 • 7.
Из канонического разложения числа легко можно вывести следующую лемму: Если n имеет вид (1), то , то все делители этого числа имеют вид:
`d = p_1^(beta_1)*p_2^(beta_2)*……*p_k^(beta^k)`, где `0 <= beta_m <= alpha_m` ( `m = 1,2,…, k`)
(2)
В самом деле, очевидно, что всякое d вида (2) делит а. Обратно, пусть d делит а, тогда a=cd, где с — некоторое натуральное число и, следовательно, все простые делители числа d входят в каноническое разложение числа а с показателями, не превышающими соответствующих показателей числа а.
Рассмотрим две функции, заданные на множестве натуральных чисел:
а) τ(n) – число всех натуральных делителей n;
2) σ(n) сумма всех натуральных делителей числа n.
Пусть n имеет каноническое разложение (1). Выведем формулы для числа и суммы его его натуральных делителей.
Теорема 1. Число натуральных делителей числа n
`tau(n) = (alpha_1 + 1)*(alpha_2 + 1)*…..*(alpha_k + 1);`
(3)
Доказательство.
читать дальше
Пример. Число 2 520 = 2³ • З² • 5 • 7. имеет (3+1)(2+1)(1+1)(1+1) = 48 делителей.
Теорема 2. Пусть n имеет каноническое разложение (1). Тогда сумма натуральных делителей числа n равна
`sigma(n) = (1 + p_1 + p_1^2 + ….. + p_1^(alpha_1))*(1 + p_2 + p_2^2 + ….. + p_2^(alpha_2))* …………..* (1 + p_k + p_k^2 + …..+ p_k^(alpha_k));`
(4)
Доказательство.
читать дальше
Пример. Найти сумму всех делителей числа 90.
90=2 • З² • 5. Тогда σ(90)=[(2²-1)/(2-1)]• [З³-1)/(3-1)]• [(5²-1)/(5-1)]=234
Формула (4) может помочь найти все делители числа.Так, например, чтобы найти все делители числа 90, раскроем скобки в следующем произведении (не производя операцию сложения): (1+2)(1+3+З²)(1+5)=(1+1*3+1*З²+1*2+2*3+2*З²)(1+5) = 1+3+З²+2+2*3+2*З²+ 5+3*5+З²*5+2*5+2*3*5+2*З²*5 = 1+3+9+2+6+18+5+15+45+10+30+90 – слагаемыми являются делители числа 90.
Решим несколько задач на тему “Число и сумма натуральных делителей натурального числа”
Задание 1. Найдите натуральное число, зная, что оно имеет только два простых делителя, что число всех делителей равно 6, а сумма всех делителей — 28.
Решение
Задания из сборника TTZ – ЕГЭ 2010. Математика. Типовые тестовые задания
Задание 2. TTZ.С6.2 Найдите все натуральные числа, которые делятся на 42 и имеют ровно 42 различных натуральных делителя (включая единицу и само число).
Решение
Задание 3. TTZ.С6.9 Найдите все натуральные числа, последняя десятичная цифра которых 0 и которые имеют ровно 15 различных натуральных делителей(включая единицу и само число).
Решение
Задание 4. SPI.С6.9. У натурального числа n ровно 6 делителей. Сумма этих делителей равна 3500. Найти n.
Решение  VEk:
Решение

Задания для самостоятельной работы
SR1. Найти все числа, имеющие ровно 2 простых делителя, всего 8 делителей, сумма которых равна 60.
SR2. Найти натуральные числа, которые делятся на 3 и на 4 и имеют ровно 21 натуральный делитель.
SR3. Найти наименьшее натуральное число, имеющее ровно 18 натуральных делителей.
SR4. Найти наименьшее число, кратное 5, имеющее 18 натуральных делителей.
SR5. Некоторое натуральное число имеет два простых делителя. Его квадрат имеет всего 15 делителей. Сколько делителей имеет куб этого числа?
SR6. Некоторое натуральное число имеет два простых делителя. Его квадрат имеет всего 81 делитель. Сколько делителей имеет куб этого числа?
SR7. Найти число вида m = 2^x3^y5^z, зная, что половина его имеет на 30 делителей меньше, треть —на 35 и пятая часть — на 42 делителя меньше, чем само число.

Топик поднят, поскольку по теме топика постоянно появляются вопросы.

Сумма делителей числа

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]]

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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]

[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]

[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]

[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]

[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 = а^х).

     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(a^qb^wc^ed^r)= σ(a^q)σ(b^w)σ(c^e)σ(d^r)

     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
≤ β_s ≤ α_s, s = 1, …, k.
Ясно, что σ(n) представляет собой сумму всех таких чисел при различных
значениях показателей

β₁,
β₂, … β_k. 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