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$${\Biggl{n:\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor^{k}\Bigr)-\gcd\Bigl(n,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor\Bigr)\gt 0}\Biggr}={\Biggl{n:\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor^{k}\Bigr)-\gcd\Bigl(n^2,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor\Bigr)\gt 0}\Biggr}={\Biggl{n:\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor^{k}\Bigr)-\gcd\Bigl(n^3,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor\Bigr)\gt 0}\Biggr}=...{\Biggl{n:\gcd\Bigl(n^k,\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor^{k}\Bigr)-\gcd\Bigl(n^{k},\Bigl\lfloor \frac{p_n}{n} \Bigr\rfloor\Bigr)\gt 0}\Biggr}$$

$\vartheta_{n,m,k}=\Biggl\lfloor \Biggl(n^{\frac{m}{k}} -\lfloor{\lfloor n^{\frac{m}{k}} \rfloor}^{\frac{k}{m}-1}\rfloor\gcd\Bigl(\lfloor{\lfloor n^{\frac{m}{k}} \rfloor}^{\frac{k}{m}-1}\rfloor,\Bigl\lfloor\Bigl\lfloor \frac{p_n^{\frac{m}{k}}}{n^{\frac{m}{k}}} \Bigr\rfloor^{\frac{k}{m}-1}\Bigr\rfloor\Bigr)\Biggr)^{\frac{k}{m}}\Biggr\rfloor$

$m \geq k \Rightarrow n-\vartheta_{n,m,k} \in {{0,1}}$

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