Let the matrix $T$ be defined by:
$$\displaystyle T(n,k) = -\varphi^{-1}(\operatorname{GCD}(n,k))$$
where $\varphi^{-1}$ is the Dirichlet inverse of the Euler totient function.
$$\varphi^{-1}(n) = \sum\limits_{d|n}\mu(d)d$$
$$\varphi^{-1}(n) = +1, -1, -2, -1, -4, +2, -6,...$$
The matrix $T$ starts:
$$\displaystyle T = -\left( \begin{array}{ccccccc} +1&+1&+1&+1&+1&+1&+1&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \\ \vdots&&&&&&&\ddots \end{array} \right)$$
Does the largest eigenvalue of the matrix $T$ approximate the previous prime number sequence?
The largest eigenvalues rounded start:
{-1, 1, 3, 3, 5, 5, 7, 7, 7, 8, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41}
compared to the previous prime number sequence:
{-2, 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41}
The agreement appears to hold for $n>10$.
Link to original question at Mathematics Stack Exchange asked 6 years ago.
Associated Mathematica program:
(*start*)
nn = 42;
A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}];
B = Table[
Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {k, 1, nn}], {n, 1,
nn}];
T = A.B;
Round[Table[
Max[Eigenvalues[
Table[Table[N[-T[[n, k]]], {k, 1, nnn}], {n, 1, nnn}]]], {nnn, 1,
nn}]]
Table[NextPrime[i, -1], {i, 2, nn+1}]
(*end*)
