Discovered a Pattern Involving Powers of Odd Primes

# https://math.stackexchange.com/questions/5021626/discovered-a-pattern-involving-powers-of-odd-primes

import sympy
from sympy import primerange
from itertools import combinations
from tabulate import tabulate

def generate_odd_primes(n):
    """
    Generate the first n odd prime numbers.
    """
    primes = list(primerange(3, sympy.prime(n) + 1))
    # Ensure we have exactly n primes
    while len(primes) < n:
        primes = list(primerange(3, sympy.prime(n) * 2))
    return primes[:n]

def find_consecutive_sum(digits, target):
    """
    Find a sequence of consecutive digits in the list `digits` that sums to `target`.
    Returns the first such sequence found.
    """
    length = len(digits)
    for window_size in range(1, length + 1):
        for i in range(length - window_size + 1):
            window = digits[i:i + window_size]
            if sum(window) == target:
                return window
    return None

def main():
    N = 10  # Number of cases to demonstrate
    primes = generate_odd_primes(N)

    table = []
    for n in range(1, N + 1):
        p_n = primes[n - 1]
        p_n_power_n = pow(p_n, n)
        digits = [int(d) for d in str(p_n_power_n)]
        sequence = find_consecutive_sum(digits, p_n)

        if sequence:
            S_pn = p_n
            sequence_str = ''.join(map(str, sequence))
            verification = f"{'+'.join(map(str, sequence))}={S_pn}"
        else:
            S_pn = "Not Found"
            sequence_str = "N/A"
            verification = "N/A"
            print(f"\n[ALERT] Sequence summing to p_n={p_n} not found in p_n^n={p_n_power_n}.\n")
            input("Press Enter to continue...")  # Pause the program

        table.append([n, p_n, p_n_power_n, verification])

    headers = ["n", "p_n", "(p_n)^n", "S(p_n)"]
    print(tabulate(table, headers, tablefmt="grid"))

if __name__ == "__main__":
    main()