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 implementation of the idea.
C++
// 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" ; } |
Java
// Java program to check whether a number is a // Sphenic number or not import java.util.*; class GFG { // create a global array of size 10001; static boolean []arr = new boolean [ 1001 ]; // This functions finds all primes smaller than 'limit' // using simple sieve of eratosthenes. static 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. Arrays.fill(arr, true ); // 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 ; } } } static int find_sphene( int N) { int []arr1 = new int [ 8 ]; // 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 < 8 ) { 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 code public static void main(String[] args) { int n = 60 ; simpleSieve(); int ans = find_sphene(n); if (ans == 1 ) System.out.print( "Yes" ); else System.out.print( "NO" ); } } // This code is contributed by aashish1995 |
Python3
# Python3 program to check whether a number # is a Sphenic number or not # Create a global array of size 1001; arr = [ True ] * ( 1001 ) # This functions finds all primes smaller # than 'limit' using simple sieve of # eratosthenes. def 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. k = 0 # One by one traverse all numbers so # that their multiples can be marked # as composite. for p in range ( 2 , 1001 ): if (p * p > 1001 ): break # If p is not changed, then it is a prime if (arr[p]): # Update all multiples of p for k in range (p, 1001 , k + p): arr[k] = False def find_sphene(N): # To store the 8 divisors arr1 = [ 0 ] * ( 8 ) # To count the number of divisor count = 0 j = 0 for i in range ( 1 , N + 1 ): if (N % i = = 0 and count < 8 ): count + = 1 arr1[j] = i j + = 1 # Finally check if there re 8 divisor and # all the numbers are distinct prime no return 1 # else return 0); if (count = = 8 and (arr[arr1[ 1 ]] and arr[arr1[ 2 ]] and arr[arr1[ 3 ]])): return 1 ; return 0 ; # Driver code if __name__ = = '__main__' : n = 60 simpleSieve() ans = find_sphene(n) if (ans = = 1 ): print ( "Yes" ) else : print ( "NO" ) # This code is contributed by gauravrajput1 |
C#
// C# program to check whether a number // is a Sphenic number or not using System; class GFG{ // Create a global array of size 10001; static bool []arr = new bool [1001]; // This functions finds all primes smaller than // 'limit'. Using simple sieve of eratosthenes. static 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. for ( int i = 0;i<1001;i++) arr[i] = true ; // 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 ; } } } static int find_sphene( int N) { // To store the 8 divisors int []arr1 = new int [8]; // To count the number of divisor int count = 0; int j = 0; for ( int i = 1; i <= N; i++) { if (N % i == 0 && count < 8) { 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 code public static void Main(String[] args) { int n = 60; simpleSieve(); int ans = find_sphene(n); if (ans == 1) Console.Write( "Yes" ); else Console.Write( "NO" ); } } // This code is contributed by aashish1995 |
Output:
NO
References:
1. OEIS
2. https://en.wikipedia.org/wiki/Sphenic_number
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