Given a natural number n, the task is to find sum of divisors of all the divisors of n.
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 = p1k1 x p2k2 x … where p1, p2, … are prime numbers.
All the divisors of n can be expressed as p1a x p2b x …, where 0 <= a <= k1 and 0 <= b <= k2.
Now sum of divisors will be sum of all power of p1 – p10, p11,…., p1k1 multiplied by all power of p2 – p20, p21,…., p2k1
Sum of Divisor of n
= (p10 x p20) + (p11 x p20) +…..+ (p1k1 x p20) +….+ (p10 x p21) + (p11 x p21) +…..+ (p1k1 x p21) +……..+
(p10 x p2k2) + (p11 x p2k2) +……+ (p1k1 x p2k2).
= (p10 + p11 +…+ p1k1) x p20 + (p10 + p11 +…+ p1k1) x p21 +…….+ (p10 + p11 +…+ p1k1) x p2k2.
= (p10 + p11 +…+ p1k1) x (p20 + p21 +…+ p2k2).
Now, the divisors of any pa, for p as prime, are p0, p1,……, pa. And sum of diviors 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(p1k1)) x (f(p20) + f(p21) +…+ f(p2k2)).
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 ab x ac = ab+c.
Below is the implementation of this approach:
This article is contributed by Anuj Chauhan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.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.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.