Squarefree binomial coefficients.

Question

At $n=23$, all binomial coefficients are squarefree. Is this the largest value for $n$ where this is the case?

Edit

A plot up to $n=50$:

A plot up to $n=500$:

plotted against $n+1$ and $\frac{112}{\sqrt{239}}\sqrt{n}$

and the same up to $n=2000$:

Do these bounds hold for all $n$?

(Clearly, the bound $n+1$ holds for all $n$.)

Update

Just out of interest, a plot up to $n=2000$ with bounding curve $24\log(n)$

($24\approx\frac{148}{\log 479}$, where $148$ is the number of squarefree binomial coefficients at $n=479$),

which seems to be the tightest curve which still holds for up to $n=3967$:

... seems to suggest that the bound is $const. \log(n)$, which so far shows $c\approx24$ (which, coincidentally, is the number of squarefree coefficients at $n=23$).

Answer 1

See here. The $n=23$ case is the last.