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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 $n=50$:

enter image description here

A plot up to $n=500$:

enter image description here

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

and the same up to $n=2000$:

enter image description here

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 $n=3967$:

enter image description here

... 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 $n=23$).

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  • $\begingroup$ I checked numerically up to $n=10^5$, and $23$ is still the largest. $\endgroup$ – Jean-Claude Arbaut Nov 28 '13 at 16:40
  • $\begingroup$ How did you check up till there? Did you use a program? $\endgroup$ – Ahaan S. Rungta Nov 28 '13 at 16:41
  • $\begingroup$ Yes. In GAP: Filtered([1..100000], n -> ForAll([0..n], k -> MoebiusMu(Binomial(n,k))<>0)); $\endgroup$ – Jean-Claude Arbaut Nov 28 '13 at 16:41
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    $\begingroup$ Actually, it's the last, see OEIS sequence A048278 $\endgroup$ – Jean-Claude Arbaut Nov 28 '13 at 16:43
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    $\begingroup$ In case one is interested, the next "record" value after n=479 (148 binomials are square free) is for n=3967 (182 square free). Checked up to 4500. $\endgroup$ – Jean-Claude Arbaut Nov 28 '13 at 22:11
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See here. The $n=23$ case is the last.

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  • $\begingroup$ @ Ahaan Rungta - many thanks for your link :) $\endgroup$ – martin Nov 28 '13 at 16:59
  • $\begingroup$ Actually, it's arbautjc's credit. See his comment. $\endgroup$ – Ahaan S. Rungta Nov 28 '13 at 17:01

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