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Edit 2017.03.06 GRP: After finding a packing for $n=65$ (leaving $ 21,39,44,$ and $55$ to handle the excess), I find that leaving out $65$ makes it impossible to produce an exact packing for $n=63$, as $13$ needs to be left out, and so do a multiple of $7$ and a multiple of $11$, which is too much for the excess over $4$ of the smooth denominators that are candidates for a good packing. So after $30$, the next $n$ to admit an exact packing are $65$ through $82$.

Edit 2017.03.06 GRP: After finding a packing for $n=65$ (leaving $ 21,39,44,$ and $55$ to handle the excess), I find that leaving out $65$ makes it impossible to produce an exact packing for $n=63$, as $13$ needs to be left out of the good subset, and so do a multiple of $7$ and a multiple of $11$, which is too much for the excess over $4$ of the smooth denominators that are candidates for a good packing. So after $30$, the next $n$ to admit an exact packing are $65$ through $82$.

Gerhard "Who Wants To Go Further?" Paseman, 2017.03.03.

Edit 2017.03.06 GRP: After finding a packing for $n=65$ (leaving $ 21,39,44,$ and $55$ to handle the excess), I find that leaving out $65$ makes it impossible to produce an exact packing for $n=63$, as $13$ needs to be left out, and so do a multiple of $7$ and a multiple of $11$, which is too much for the excess over $4$ of the smooth denominators that are candidates for a good packing. So after $30$, the next $n$ to admit an exact packing are $65$ through $82$.

Based on gut feel and some trial nomogramming, I suspect the next lowest admissible $n$ will be around $170$ or greater. So far the solutions above are found by hand. The exploration continues. End Edit 2017.03.06 GRP.

Gerhard "Who Wants To Go Further?" Paseman, 2017.03.03.

2  4*    8*              16*
  3   6    9*   12          18        24     27*
     5       10        15        20        25*        30
         7            14               21              28

2  4*    8*              16*
  3   6    9*   12          18        24     27*
     5       10        15        20        25*        30
         7            14             21             28