looking for patterns in primes as co_p_factor numbers

primes_patterns.py

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+import math
+
+def primes_up_to(n: int):
+    """Generate all primes ≤ n using sieve of Eratosthenes."""
+    if n < 2:
+        return []
+    sieve = [True] * (n + 1)
+    sieve[0] = sieve[1] = False
+    for p in range(2, int(math.isqrt(n)) + 1):
+        if sieve[p]:
+            step = p
+            start = p * p
+            sieve[start:n + 1:step] = [False] * ((n - start) // step + 1)
+    return [i for i, is_p in enumerate(sieve) if is_p]
+
+def factor_exponents(n: int):
+    """Return dict {prime: exponent} for n (n ≥ 1)."""
+    exps = {}
+    if n <= 1:
+        return exps
+    limit = math.isqrt(n)
+    for p in primes_up_to(limit):
+        if p * p > n:
+            break
+        while n % p == 0:
+            exps[p] = exps.get(p, 0) + 1
+            n //= p
+    if n > 1:  # leftover prime
+        exps[n] = exps.get(n, 0) + 1
+    return exps
+
+def exponents_in_prime_order(n: int, min_len=7):
+    """
+    Produce exponents in order [2, 3, 5, 7, 11, 13, ...],
+    padded with zeros to length >= min_len.
+    """
+    if n == 1:
+        return [0] * min_len
+
+    exps = factor_exponents(n)
+
+    # generate primes in ascending order until we cover all factors
+    primes_needed = sorted(exps.keys())
+    max_prime = max(primes_needed) if primes_needed else 2
+    primes_list = primes_up_to(max_prime)
+
+    out = [exps.get(p, 0) for p in primes_list]
+
+    # pad with zeros
+    while len(out) < min_len:
+        out.append(0)
+
+    return out
+
+def format_exponents(exps):
+    """Format as comma-separated list with zero-padded width=3."""
+    return ",".join(f"{e:03d}" for e in exps)
+
+def main():
+  for s in range(1,100):
+    #s = input("Enter a positive integer (1..10000000): ").strip()
+    try:
+        n = int(s)
+    except ValueError:
+        print("Error: please enter an integer.")
+        return
+    if n < 1 or n > 10_000_000:
+        print("Error: number must be between 1 and 10,000,000.")
+        return
+
+    exps = exponents_in_prime_order(n, min_len=7)
+    print(s, ': ',format_exponents(exps))
+
+if __name__ == "__main__":
+    main()
+