给定两个整数X和Y ,任务是执行以下操作:
- 找出[X,Y]范围内的所有素数。
- 通过组合给定范围内的每对素数生成所有可能的数字。
- 在上面生成的所有可能的数字中找到质数。计算其中素数的数量,以N表示。
- 打印斐波纳契数列的第N个项,该数列是由上述列表中最小和最大的素数组成的,作为该序列的前两个项。
例子:
Input: X = 2 Y = 40
Output: 34
Explanation:
All primes in the range [X, Y] = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
All possible numbers generated by concatenating each pair of prime = [23, 25, 27, 211, 213, 217, 219, 223, 229, 231, 32, 35, 37, 311, 313, 319, 323, 329, 331, 337, 52, 53, 57, 511, 513, 517, 519, 523, 529, 531, 537, 72, 73, 75, 711, 713, 717, 719, 723, 729, 731, 737, 112, 113, 115, 117, 1113, 1117, 1119, 1123, 1129, 1131, 1137, 132, 133, 135, 137, 1311, 1317, 1319, 1323, 1329, 1331, 1337, 172, 173, 175, 177, 1711, 1713, 1719, 1723, 1729, 1731, 1737, 192, 193, 195, 197, 1911, 1913, 1917, 1923, 1929, 1931, 1937, 232, 233, 235, 237, 2311, 2313, 2317, 2319, 2329, 2331, 2337, 292, 293, 295, 297, 2911, 2913, 2917, 2919, 2923, 2931, 2937, 312, 315, 317, 3111, 3113, 3117, 3119, 3123, 3129, 3137, 372, 373, 375, 377, 3711, 3713, 3717, 3719, 3723, 3729, 3731]
All primes among the generated numbers=[193, 3137, 197, 2311, 3719, 73, 137, 331, 523, 1931, 719, 337, 211, 23, 1117, 223, 1123, 229, 37, 293, 2917, 1319, 1129, 233, 173, 3119, 113, 53, 373, 311, 313, 1913, 1723, 317]
Count of the primes = 34
Smallest Prime = 23
Largest Prime = 3719
Therefore, the 34th term of the Fibonacci series having 23 and 3719 as the first two terms, is 13158006689.
Input: X = 1, Y = 10
Output: 1053
方法:
请按照以下步骤解决问题:
- 使用Eratothenes筛网生成所有可能的质数。
- 遍历范围[X,Y]并借助上一步中生成的primes []数组生成该范围内的所有素数。
- 遍历素数列表并从列表中生成所有可能的对。
- 对于每对,将两个素数连接起来,然后检查它们的串联是否为素数。
- 找出所有此类素数的最大值和最小值,并对获得的所有此类素数进行计数。
- 最后,将在上述步骤中获得的具有最小值和最大值的斐波那契数列的计数th打印为该数列的前两项。
下面是上述方法的实现:
C++
// C++ program to implement
// the above approach
#include
using namespace std;
#define int long long int
// Stores at each index if it's a
// prime or not
int prime[100001];
// Sieve of Eratosthenes to
// generate all possible primes
void SieveOfEratosthenes()
{
for(int i = 0; i < 100001; i++)
prime[i] = 1;
int p = 2;
while (p * p <= 100000)
{
// If p is a prime
if (prime[p] == 1)
{
// Set all multiples of
// p as non-prime
for(int i = p * p;
i < 100001;
i += p)
prime[i] = 0;
}
p += 1;
}
}
int join(int a, int b)
{
int mul = 1;
int bb = b;
while(b != 0)
{
mul *= 10;
b /= 10;
}
a *= mul;
a += bb;
return a;
}
// Function to generate the
// required Fibonacci Series
void fibonacciOfPrime(int n1, int n2)
{
SieveOfEratosthenes();
// Stores all primes between
// n1 and n2
vectorinitial;
// Generate all primes between
// n1 and n2
for(int i = n1; i <= n2; i++)
if (prime[i])
initial.push_back(i);
// Stores all concatenations
// of each pair of primes
vectornow;
// Generate all concatenations
// of each pair of primes
for(auto a:initial)
for(auto b:initial)
if (a != b)
{
int c = join(a,b);
now.push_back(c);
}
// Stores the primes out of the
// numbers generated above
vectorcurrent;
for(auto x:now)
if (prime[x])
current.push_back(x);
// Store the unique primes
setcurrent_set;
for(auto i:current)
current_set.insert(i);
// Find the minimum
int first = *min_element(current_set.begin(),
current_set.end());
// Find the minimum
int second = *max_element(current_set.begin(),
current_set.end());
// Find N
int count = current_set.size() - 1;
int curr = 1;
int c;
while( curr < count)
{
c = first + second;
first = second;
second = c;
curr += 1;
}
// Print the N-th term
// of the Fibonacci Series
cout << (c) << endl;
}
// Driver Code
int32_t main()
{
int x = 2;
int y = 40;
fibonacciOfPrime(x, y);
}
// This code is contributed by Stream_Cipher
Java
// Java program to implement
// the above approach
import java.util.*;
class GFG{
// Stores at each index if it's a
// prime or not
static int prime[] = new int [100001];
// Sieve of Eratosthenes to
// generate all possible primes
static void SieveOfEratosthenes()
{
for(int i = 0; i < 100001; i++)
prime[i] = 1;
int p = 2;
while (p * p <= 100000)
{
// If p is a prime
if (prime[p] == 1)
{
// Set all multiples of
// p as non-prime
for(int i = p * p;
i < 100001;
i += p)
prime[i] = 0;
}
p += 1;
}
}
static int join(int a,int b)
{
int mul = 1;
int bb = b;
while(b != 0)
{
mul *= 10;
b /= 10;
}
a *= mul;
a += bb;
return a;
}
// Function to generate the
// required Fibonacci Series
static void fibonacciOfPrime(int n1, int n2)
{
SieveOfEratosthenes();
// Stores all primes between
// n1 and n2
Vector initial = new Vector<>();
// Generate all primes between
// n1 and n2
for(int i = n1; i <= n2; i++)
if (prime[i] == 1)
initial.add(i);
// Stores all concatenations
// of each pair of primes
Vector now = new Vector<>();
// Generate all concatenations
// of each pair of primes
for(int i = 0; i < initial.size(); i++)
{
for(int j = 0; j < initial.size(); j++)
{
int a = (int)initial.get(i);
int b = (int)initial.get(j);
if (a != b)
{
int c = join(a, b);
now.add(c);
}
}
}
// Stores the primes out of the
// numbers generated above
Vector current = new Vector<>();
for(int i = 0; i < now.size(); i++)
if (prime[(int)now.get(i)] == 1)
current.add((int)now.get(i));
// Store the unique primes
int cnt[] = new int[1000009];
for(int i = 0; i < 1000001; i++)
cnt[i] = 0;
Vector current_set = new Vector<>();
for(int i = 0; i < current.size(); i++)
{
cnt[(int)current.get(i)]++;
if (cnt[(int)current.get(i)] == 1)
current_set.add((int)current.get(i));
}
// Find the minimum
long first = 1000000000;
for(int i = 0; i < current_set.size(); i++)
first = Math.min(first,
(int)current_set.get(i));
// Find the minimum
long second = 0;
for(int i = 0; i < current_set.size(); i++)
second = Math.max(second,
(int)current_set.get(i));
// Find N
int count = current_set.size() - 1;
long curr = 1;
long c = 0;
while(curr < count)
{
c = first + second;
first = second;
second = c;
curr += 1;
}
// Print the N-th term
// of the Fibonacci Series
System.out.println(c);
}
// Driver code
public static void main(String[] args)
{
int x = 2;
int y = 40;
fibonacciOfPrime(x, y);
}
}
// This code is contributed by Stream_Cipher
Python3
# Python3 Program to implement
# the above approach
# Stores at each index if it's a
# prime or not
prime = [True for i in range(100001)]
# Sieve of Eratosthenes to
# generate all possible primes
def SieveOfEratosthenes():
p = 2
while (p * p <= 100000):
# If p is a prime
if (prime[p] == True):
# Set all multiples of p as non-prime
for i in range(p * p, 100001, p):
prime[i] = False
p += 1
# Function to generate the
# required Fibonacci Series
def fibonacciOfPrime(n1, n2):
SieveOfEratosthenes()
# Stores all primes between
# n1 and n2
initial = []
# Generate all primes between
# n1 and n2
for i in range(n1, n2):
if prime[i]:
initial.append(i)
# Stores all concatenations
# of each pair of primes
now = []
# Generate all concatenations
# of each pair of primes
for a in initial:
for b in initial:
if a != b:
c = str(a) + str(b)
now.append(int(c))
# Stores the primes out of the
# numbers generated above
current = []
for x in now:
if prime[x]:
current.append(x)
# Store the unique primes
current = set(current)
# Find the minimum
first = min(current)
# Find the minimum
second = max(current)
# Find N
count = len(current) - 1
curr = 1
while curr < count:
c = first + second
first = second
second = c
curr += 1
# Print the N-th term
# of the Fibonacci Series
print(c)
# Driver Code
if __name__ == "__main__":
x = 2
y = 40
fibonacciOfPrime(x, y)
C#
// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
class GFG{
// Stores at each index if it's a
// prime or not
static int[] prime = new int[100001];
// Sieve of Eratosthenes to
// generate all possible primes
static void SieveOfEratosthenes()
{
for(int i = 0; i < 100001; i++)
prime[i] = 1;
int p = 2;
while (p * p <= 100000)
{
// If p is a prime
if (prime[p] == 1)
{
// Set all multiples of
// p as non-prime
for(int i = p * p;
i < 100001; i += p)
prime[i] = 0;
}
p += 1;
}
}
static int join(int a,
int b)
{
int mul = 1;
int bb = b;
while(b != 0)
{
mul *= 10;
b /= 10;
}
a *= mul;
a += bb;
return a;
}
// Function to generate the
// required Fibonacci Series
static void fibonacciOfPrime(int n1,
int n2)
{
SieveOfEratosthenes();
// Stores all primes
// between n1 and n2
List initial =
new List();
// Generate all primes
// between n1 and n2
for(int i = n1; i <= n2; i++)
if (prime[i] == 1)
initial.Add(i);
// Stores all concatenations
// of each pair of primes
List now =
new List();
// Generate all concatenations
// of each pair of primes
for(int i = 0; i < initial.Count; i++)
{
for(int j = 0; j < initial.Count; j++)
{
int a = initial[i];
int b = initial[j];
if (a != b)
{
int C = join(a, b);
now.Add(C);
}
}
}
// Stores the primes out of the
// numbers generated above
List current =
new List();
for(int i = 0; i < now.Count; i++)
if (prime[now[i]] == 1)
current.Add(now[i]);
// Store the unique primes
int[] cnt = new int[1000009];
for(int i = 0; i < 1000001; i++)
cnt[i] = 0;
List current_set =
new List();
for(int i = 0; i < current.Count; i++)
{
cnt[current[i]]++;
if (cnt[current[i]] == 1)
current_set.Add(current[i]);
}
// Find the minimum
long first = 1000000000;
for(int i = 0;
i < current_set.Count; i++)
first = Math.Min(first,
current_set[i]);
// Find the minimum
long second = 0;
for(int i = 0;
i < current_set.Count; i++)
second = Math.Max(second,
current_set[i]);
// Find N
int count = current_set.Count - 1;
long curr = 1;
long c = 0;
while(curr < count)
{
c = first + second;
first = second;
second = c;
curr += 1;
}
// Print the N-th term
// of the Fibonacci Series
Console.WriteLine(c);
}
// Driver code
static void Main()
{
int x = 2;
int y = 40;
fibonacciOfPrime(x, y);
}
}
// This code is contributed by divyeshrabadiya07
13158006689
时间复杂度: O(N 2 + log(log(maxm))),其中O(N 2 )生成所有对,O(1)检查一个数字是否为质数,maxm为质数的大小[]
辅助空间: O(maxm)