A Sphenic Number is a positive integer n which is product of exactly three distinct primes. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, …
Given a number n, determine whether it is a Sphenic Number or not.
Examples:
Input : 30
Output : Yes
Explanation : 30 is the smallest Sphenic number,
30 = 2 × 3 × 5
the product of the smallest three primes
Input : 60
Output : No
Explanation : 60 = 22 x 3 x 5
has exactly 3 prime factors but
is not a sphenic number
Sphenic number can be checked by fact that every sphenic number will have exactly 8 divisor SPHENIC NUMBER
So first We will try to find if the number is having exactly 8 divisors if not then simply answer is no.If there are exactly 8 divisors then we will confirm weather the first 3 digits after 1 are prime or not.
Eg. 30 (sphenic number)
30=p*q*r(i.e p,q and r are three distinct prime no and their product are 30)
the set of divisor is (1,2,3,5,6,10,15,30).
Below is the C++ implementation of the idea.
// C++ program to check whether a number is a // Sphenic number or not #include<bits/stdc++.h> using namespace std; //create a global array of size 10001; bool arr[1001]; // This functions finds all primes smaller than 'limit' // using simple sieve of eratosthenes. void simpleSieve() { // initialize all entries of it as true. A value // in mark[p] will finally be false if 'p' is Not // a prime, else true. memset(arr,true,sizeof(arr)); // One by one traverse all numbers so that their // multiples can be marked as composite. for(int p=2;p*p<1001;p++) { // If p is not changed, then it is a prime if(arr[p]) {// Update all multiples of p for(int i=p*2;i<1001;i=i+p) arr[i]=false; } } } int find_sphene(int N) { int arr1[8]={0}; //to store the 8 divisors int count=0; //to count the number of divisor int j=0; for(int i=1;i<=N;i++) { if(N%i==0 &&count<9) { count++; arr1[j++]=i; } } //finally check if there re 8 divisor and all the numbers are distinct prime no return 1 //else return 0 if(count==8 && (arr[arr1[1]] && arr[arr1[2]] && arr[arr1[3]])) return 1; return 0; } // Driver program to test above function int main() { int n = 60; simpleSieve(); int ans=find_sphene(n); if(ans) cout<<"Yes"; else cout<<"NO"; } |
Output:
Yes
References:
1. OEIS
2. https://en.wikipedia.org/wiki/Sphenic_number
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Improved By : Mohammedansari1111145
