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I posted this question at the Mathematics SE, but received no response there so I am posting it here.

The following fact is stated in the comments-section of sequence A013928 in the OEIS.

Let $C$ be the $n$-by-$n$ matrix whose $(i,j)$-entry is one if $\gcd(i,j)=1$ and zero otherwise.

It is claimed that $\text{rank}(C) = Q(n+1)$, where $Q(n)$ denotes the number of squarefree integers less than $n$.

Looking for a reference on this claim (proving the claim is not an issue) and whether this matrix has been studied previously.

Improve this question
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  • $\begingroup$ Have you looked at the links and references for that sequence? $\endgroup$
    – Robert Israel
    Jul 6, 2018 at 0:22
  • $\begingroup$ @RobertIsrael: I did. That claim and matrix do not appear in any of the references. $\endgroup$
    – Pietro Paparella
    Jul 6, 2018 at 0:24
  • $\begingroup$ I don't know if this matrix appears in the literature, although it is likely that it does. Clements and Lindstroem have a slightly different binary matrix with large determinant. It is possible that there is an article which references both matrices, in which case start with a citation search on the Clements and Lindstroem article. Gerhard "Use High Rank For Low" Paseman, 2018.07.05. $\endgroup$
    – Gerhard Paseman
    Jul 6, 2018 at 3:01
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    $\begingroup$ The fact $rank(C)=Q(n+1)$ can be easily proved, just notice that $gcd(x,p_{1}^{k_1}p_{2}^{k_2}...p_{n}^{k_n})=1 \Leftrightarrow gcd(x,p_{1}p_{2}...p_{n})=1$, and the squarefree columns of the matrix form a basis. $\endgroup$
    – LeechLattice
    Jul 6, 2018 at 5:56

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