Super-prime numbers (also known as higher order primes) are the subsequence of prime number sequence that occupy prime-numbered positions within the sequence of all prime numbers. The first few super primes are 3, 5, 11, and 17.
Given a positive integer n and the task is to print all the Super-Primes less than or equal to n.
Examples:Â
Input: n = 5
Output: [3, 5]
Explanation: The prime numbers up to 5 are [2, 3, 5]. Their positions are [1, 2, 3]. Since positions 2 and 3 are prime, 3 and 5 are super-primes.Input: n = 20
Output: [3, 5, 11, 17]
Explanation: The prime numbers up to 20 are [2, 3, 5, 7, 11, 13, 17, 19]. Their positions are [1, 2, 3, 4, 5, 6, 7, 8]. Since positions 2, 3, 5, and 7 are prime, 3, 5, 11, and 17 are super-primes.
Try It Yourself
Table of Content
[Naive Approach] Check Primality Individually - O(n√n) Time and O(n) Space
The idea is to find all prime numbers up to n and store them.
Then, check whether the 1-based position of each prime is also prime. If yes, add that prime to the result.
Working of Approach:
- Traverse all numbers from 2 to n.
- Check each number individually to determine whether it is prime.
- Store all prime numbers in a vector.
- Traverse the vector and check whether each prime's 1-based position is prime.
- Add the corresponding prime number to the result.
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
// Function to check whether a number is prime.
bool isPrime(int num)
{
if (num < 2)
return false;
for (int i = 2; i * i <= num; i++)
{
if (num % i == 0)
return false;
}
return true;
}
vector<int> superPrimes(int n)
{
vector<int> primes;
vector<int> res;
// Find and store all prime numbers up to n.
for (int i = 2; i <= n; i++)
{
if (isPrime(i))
primes.push_back(i);
}
// Check whether the position of each prime is also prime.
for (int i = 0; i < primes.size(); i++)
{
if (isPrime(i + 1))
res.push_back(primes[i]);
}
return res;
}
int main()
{
int n = 20;
vector<int> ans = superPrimes(n);
cout << "[";
for (int i = 0; i < ans.size(); i++)
{
cout << ans[i];
if (i != ans.size() - 1)
cout << ", ";
}
cout << "]";
return 0;
}
import java.util.ArrayList;
class GFG {
// Function to check whether a number is prime.
public boolean isPrime(int num)
{
if (num < 2)
return false;
for (int i = 2; i * i <= num; i++) {
if (num % i == 0)
return false;
}
return true;
}
public ArrayList<Integer> superPrimes(int n)
{
ArrayList<Integer> primes = new ArrayList<>();
ArrayList<Integer> res = new ArrayList<>();
// Find and store all prime numbers up to n.
for (int i = 2; i <= n; i++) {
if (isPrime(i))
primes.add(i);
}
// Check whether the position of each prime is also
// prime.
for (int i = 0; i < primes.size(); i++) {
if (isPrime(i + 1))
res.add(primes.get(i));
}
return res;
}
public static void main(String[] args)
{
int n = 20;
GFG ob = new GFG();
ArrayList<Integer> ans = ob.superPrimes(n);
System.out.println(ans);
}
}
# Function to check whether a number is prime.
def isPrime(num):
if num < 2:
return False
for i in range(2, int(num ** 0.5) + 1):
if num % i == 0:
return False
return True
def superPrimes(n):
primes = []
res = []
# Find and store all prime numbers up to n.
for i in range(2, n + 1):
if isPrime(i):
primes.append(i)
# Check whether the position of each prime is also prime.
for i in range(len(primes)):
if isPrime(i + 1):
res.append(primes[i])
return res
if __name__ == '__main__':
n = 20
ans = superPrimes(n)
print('[', end='')
for i in range(len(ans)):
print(ans[i], end='' if i == len(ans) - 1 else ', ')
print(']')
using System;
using System.Collections.Generic;
// Function to check whether a number is prime.
public class GFG {
public static bool IsPrime(int num)
{
if (num < 2)
return false;
for (int i = 2; i * i <= num; i++) {
if (num % i == 0)
return false;
}
return true;
}
public static List<int> superPrimes(int n)
{
List<int> primes = new List<int>();
List<int> res = new List<int>();
// Find and store all prime numbers up to n.
for (int i = 2; i <= n; i++) {
if (IsPrime(i))
primes.Add(i);
}
// Check whether the position of each prime is also
// prime.
for (int i = 0; i < primes.Count; i++) {
if (IsPrime(i + 1))
res.Add(primes[i]);
}
return res;
}
public static void Main()
{
int n = 20;
List<int> ans = superPrimes(n);
Console.Write("[");
for (int i = 0; i < ans.Count; i++) {
Console.Write(ans[i]);
if (i != ans.Count - 1)
Console.Write(", ");
}
Console.Write("]");
}
}
// Function to check whether a number is prime.
function isPrime(num)
{
if (num < 2)
return false;
for (let i = 2; i * i <= num; i++) {
if (num % i === 0)
return false;
}
return true;
}
function superPrimes(n)
{
let primes = [];
let res = [];
// Find and store all prime numbers up to n.
for (let i = 2; i <= n; i++) {
if (isPrime(i))
primes.push(i);
}
// Check whether the position of each prime is also
// prime.
for (let i = 0; i < primes.length; i++) {
if (isPrime(i + 1))
res.push(primes[i]);
}
return res;
}
// Driver Code
let n = 20;
let ans = superPrimes(n);
console.log("[");
for (let i = 0; i < ans.length; i++) {
process.stdout.write(ans[i].toString());
if (i !== ans.length - 1)
process.stdout.write(", ");
}
console.log("]");
Output
[3, 5, 11, 17]
[Expected Approach] Using Sieve of Eratosthenes - O(n log(log n)) Time and O(n) Space
The idea is to generate all prime numbers less than or equal to n using the Sieve of Eratosthenes.
Then, check which prime numbers occupy prime-numbered positions and add them to the result.
Working of Approach:
- Use the Sieve of Eratosthenes to mark all prime numbers up to n.
- Store all the generated prime numbers in an array.
- Traverse the array of prime numbers.
- For each prime at index k, check whether k + 1 is prime.
- If its position is prime, add the number to the result.
Let us understand with an example:
- For n = 20, the Sieve of Eratosthenes finds the prime numbers: [2, 3, 5, 7, 11, 13, 17, 19].
- Their 1-based positions are 1, 2, 3, 4, 5, 6, 7, 8.
- Prime positions among them are 2, 3, 5, and 7.
- Therefore, the prime numbers at these positions are 3, 5, 11, and 17.
- Hence, the output is [3, 5, 11, 17].
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
// Generate all prime numbers less than n.
void sieveOfEratosthenes(int n, bool isPrime[])
{
// Initialize all entries of boolean array as true. A
// value in isPrime[i] will finally be false if i is Not
// a prime, else true bool isPrime[n+1];
isPrime[0] = isPrime[1] = false;
for (int i = 2; i <= n; i++)
isPrime[i] = true;
for (int p = 2; p * p <= n; p++)
{
// If isPrime[p] is not changed, then it is a prime
if (isPrime[p] == true)
{
// Update all multiples of p
for (int i = p * 2; i <= n; i += p)
isPrime[i] = false;
}
}
}
vector<int> superPrimes(int n)
{
// Generating primes using Sieve
vector<int> res;
bool isPrime[n + 1];
sieveOfEratosthenes(n, isPrime);
// Storing all the primes generated in a an array
// primes[]
int primes[n + 1], j = 0;
for (int p = 2; p <= n; p++)
if (isPrime[p])
primes[j++] = p;
// Printing all those prime numbers that occupy prime
// numbered position in sequence of prime numbers.
for (int k = 0; k < j; k++)
if (isPrime[k + 1])
res.push_back(primes[k]);
return res;
}
int main()
{
int n = 20;
vector<int> ans = superPrimes(n);
cout << "[";
for (int i = 0; i < ans.size(); i++)
{
cout << ans[i];
if (i != ans.size() - 1)
cout << ", ";
}
cout << "]";
return 0;
}
import java.util.ArrayList;
import java.util.Arrays;
public class GFG {
// Generate all prime numbers less than n.
static void sieveOfEratosthenes(int n,
boolean isPrime[])
{
// Initialize all entries of boolean array as true.
// A value in isPrime[i] will finally be false if i
// is Not a prime, else true
isPrime[0] = isPrime[1] = false;
for (int i = 2; i <= n; i++)
isPrime[i] = true;
for (int p = 2; p * p <= n; p++) {
// If isPrime[p] is not changed, then it is a
// prime
if (isPrime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= n; i += p)
isPrime[i] = false;
}
}
}
static ArrayList<Integer> superPrimes(int n)
{
// Generating primes using Sieve
ArrayList<Integer> res = new ArrayList<>();
boolean isPrime[] = new boolean[n + 1];
sieveOfEratosthenes(n, isPrime);
// Storing all the primes generated in a an array
// primes[]
int primes[] = new int[n + 1];
int j = 0;
for (int p = 2; p <= n; p++)
if (isPrime[p])
primes[j++] = p;
// Printing all those prime numbers that occupy
// prime numbered position in sequence of prime
// numbers.
for (int k = 0; k < j; k++)
if (isPrime[k + 1])
res.add(primes[k]);
return res;
}
public static void main(String[] args)
{
int n = 20;
ArrayList<Integer> ans = superPrimes(n);
System.out.print("[");
for (int i = 0; i < ans.size(); i++) {
System.out.print(ans.get(i));
if (i != ans.size() - 1)
System.out.print(", ");
}
System.out.print("]");
}
}
def sieveOfEratosthenes(n, isPrime):
# Initialize all entries of boolean array as true. A
# value in isPrime[i] will finally be false if i is Not
# a prime, else true
isPrime[0] = isPrime[1] = False
for i in range(2, n + 1):
isPrime[i] = True
for p in range(2, int(n**0.5) + 1):
# If isPrime[p] is not changed, then it is a prime
if isPrime[p] == True:
# Update all multiples of p
for i in range(p * 2, n + 1, p):
isPrime[i] = False
def superPrimes(n):
# Generating primes using Sieve
res = []
isPrime = [False] * (n + 1)
sieveOfEratosthenes(n, isPrime)
# Storing all the primes generated in a list primes[]
primes = [0] * (n + 1)
j = 0
for p in range(2, n + 1):
if isPrime[p]:
primes[j] = p
j += 1
# Printing all those prime numbers that occupy prime
# numbered position in sequence of prime numbers.
for k in range(j):
if isPrime[k + 1]:
res.append(primes[k])
return res
if __name__ == "__main__":
n = 20
ans = superPrimes(n)
print(ans)
using System;
using System.Collections.Generic;
public class GFG {
// Generate all prime numbers less than n.
public static void sieveOfEratosthenes(int n,
bool[] isPrime)
{
// Initialize all entries of boolean array as true.
// A value in isPrime[i] will finally be false if i
// is Not a prime, else true
isPrime[0] = isPrime[1] = false;
for (int i = 2; i <= n; i++)
isPrime[i] = true;
for (int p = 2; p * p <= n; p++) {
// If isPrime[p] is not changed, then it is a
// prime
if (isPrime[p] == true) {
// Update all multiples of p
for (int i = p * 2; i <= n; i += p)
isPrime[i] = false;
}
}
}
public static List<int> superPrimes(int n)
{
// Generating primes using Sieve
List<int> res = new List<int>();
bool[] isPrime = new bool[n + 1];
sieveOfEratosthenes(n, isPrime);
// Storing all the primes generated in a list
// primes[]
int[] primes = new int[n + 1];
int j = 0;
for (int p = 2; p <= n; p++)
if (isPrime[p])
primes[j++] = p;
// Printing all those prime numbers that occupy
// prime numbered position in sequence of prime
// numbers.
for (int k = 0; k < j; k++)
if (isPrime[k + 1])
res.Add(primes[k]);
return res;
}
public static void Main()
{
int n = 20;
List<int> ans = superPrimes(n);
Console.Write("[");
for (int i = 0; i < ans.Count; i++) {
Console.Write(ans[i]);
if (i != ans.Count - 1)
Console.Write(", ");
}
Console.Write("]");
}
}
function sieveOfEratosthenes(n, isPrime)
{
// Initialize all entries of boolean array as true. A
// value in isPrime[i] will finally be false if i is Not
// a prime, else true
isPrime[0] = isPrime[1] = false;
for (let i = 2; i <= n; i++)
isPrime[i] = true;
for (let p = 2; p * p <= n; p++) {
// If isPrime[p] is not changed, then it is a prime
if (isPrime[p] == true) {
// Update all multiples of p
for (let i = p * 2; i <= n; i += p)
isPrime[i] = false;
}
}
}
function superPrimes(n)
{
// Generating primes using Sieve
let res = [];
let isPrime = Array(n + 1).fill(false);
sieveOfEratosthenes(n, isPrime);
// Storing all the primes generated in a array primes[]
let primes = Array(n + 1).fill(0);
let j = 0;
for (let p = 2; p <= n; p++)
if (isPrime[p])
primes[j++] = p;
// Printing all those prime numbers that occupy prime
// numbered position in sequence of prime numbers.
for (let k = 0; k < j; k++)
if (isPrime[k + 1])
res.push(primes[k]);
return res;
}
// Driver Code
let n = 20;
let ans = superPrimes(n);
console.log("[" + ans.join(", ") + "]");
Output
[3, 5, 11, 17]
