Step by Step solution about How to resolve the algorithm Fortunate numbers step by step in the Python programming language

The given Python program finds and prints the first 50 fortunate numbers. A fortunate number is a positive integer that, when added to the primorial of its position in the sequence of positive integers, produces a prime number.

Key Concepts:

• Primorial: The primorial of a natural number n is the product of the first n prime numbers.

Function fortunate_number:

• This function takes a positive integer n as input and returns the fortunate number for that position.
• It begins by observing that the primorial of any positive integer is even. This eliminates even integers as potential fortunate numbers.
• It then starts searching for fortunate numbers from the odd integer 3.
• It calculates the primorial of n, denoted as primorial_, and iterates over odd integers, starting from 3.
• For each odd integer i, it checks if (primorial_ + i) is a prime number using sympy's isprime function. If it is prime, it returns i as the fortunate number for position n.

Main Program:

• The program initializes an empty set called fortunate_numbers to store the fortunate numbers.
• It iterates from 1 to 75, calling the fortunate_number function for each value of i and adding the result to the set.
• The program extracts the first 50 numbers from the set by sorting them and slicing the list.
• Finally, it prints the first 50 fortunate numbers in a tabular format, with 10 numbers per row.

from sympy.ntheory.generate import primorial
from sympy.ntheory import isprime

def fortunate_number(n):
    '''Return the fortunate number for positive integer n.'''
    # Since primorial(n) is even for all positive integers n,
    # it suffices to search for the fortunate numbers among odd integers.
    i = 3
    primorial_ = primorial(n)
    while True:
        if isprime(primorial_ + i):
            return i
        i += 2

fortunate_numbers = set()
for i in range(1, 76):
    fortunate_numbers.add(fortunate_number(i))

# Extract the first 50 numbers.
first50 = sorted(list(fortunate_numbers))[:50]

print('The first 50 fortunate numbers:')
print(('{:<3} ' * 10).format(*(first50[:10])))
print(('{:<3} ' * 10).format(*(first50[10:20])))
print(('{:<3} ' * 10).format(*(first50[20:30])))
print(('{:<3} ' * 10).format(*(first50[30:40])))
print(('{:<3} ' * 10).format(*(first50[40:])))