Fonksyon pi, ki konte kantite nonm premye

Ekspresyon Fonksyon ki konte kantite nonm premye selon Lainé Jean Lhermite Junior

$\pi \left(n\right)=\sum _{m=1}^{n}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor }{\frac {\left(m!\right)^{2}}{m^{3}}}}\right\rfloor \right)}=n-\sum _{m=1}^{n}{\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor }{\frac {\left(m!\right)^{2}}{m^{3}}}}\right\rfloor }=\sum _{m=1}^{n}-\left\lfloor \left\lfloor {\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor -{\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor =\sum _{m=1}^{n}\left|\left\lfloor \left\lfloor {\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor -{\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor \right|$

$\pi \left(1\right)=0$
et
$\pi \left(n\right)-\pi \left(n-1\right)={\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor }{\frac {\left(n!\right)^{2}}{n^{3}}}}\right\rfloor \right)}$

$\pi \left(1\right)=0$
et

$\pi \left(n\right)-\pi \left(n-1\right)=-\left\lfloor \left\lfloor {\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor -{\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor =\left|\left\lfloor \left\lfloor {\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor -{\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor \right|$

$\pi \left(1\right)=0$
et

$\pi \left(n-1\right)-\pi \left(n\right)=\left\lfloor \left\lfloor {\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor -{\frac {\left(n!\right)^{2}}{n^{3}}}\right\rfloor$

Gade tou

Referans

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