Python Program for Number of elements with odd factors in given range
Given a range [n,m], find the number of elements that have odd number of factors in the given range (n and m inclusive).
Input : n = 5, m = 100 Output : 8 The numbers with odd factors are 9, 16, 25, 36, 49, 64, 81 and 100 Input : n = 8, m = 65 Output : 6 Input : n = 10, m = 23500 Output : 150
A Simple Solution is to loop through all numbers starting from n. For every number, check if it has an even number of factors. If it has an even number of factors then increment count of such numbers and finally print the number of such elements. To find all divisors of a natural number efficiently, refer All divisors of a natural number
An Efficient Solution is to observe the pattern. Only those numbers, which are perfect Squares have an odd number of factors. Let us analyze this pattern through an example.
For example, 9 has odd number of factors, 1, 3 and 9. 16 also has odd number of factors, 1, 2, 4, 8, 16. The reason for this is, for numbers other than perfect squares, all factors are in the form of pairs, but for perfect squares, one factor is single and makes the total as odd.
How to find number of perfect squares in a range?
The answer is difference between square root of m and n-1 (not n)
There is a little caveat. As both n and m are inclusive, if n is a perfect square, we will get an answer which is less than one the actual answer. To understand this, consider range [4, 36]. Answer is 5 i.e., numbers 4, 9, 16, 25 and 36.
But if we do (36**0.5) – (4**0.5) we get 4. So to avoid this semantic error, we take n-1.
Count is 8
Time Complexity: O(1)
Please refer complete article on Number of elements with odd factors in given range for more details!