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I found the following formula in another math group:

$$\large\color{blue}{\pi\left( \dfrac{\left( \pi!\right)!-\lceil \pi \rceil \pi! }{{\pi}^{\sqrt \pi}-\pi!}\right)=2020}$$

It actually looks very "elegant". But, then I used Wolfram because I was in doubt.. The result shows that this formula is wrong.

I wrote these steps:

$${\qquad \quad \color{red}{\pi\left( \dfrac{\left( \pi!\right)!-\lceil \pi \rceil \pi! }{{\pi}^{\sqrt \pi}-\pi!}\right)=\color{blue}{\dfrac {\pi \Gamma(1 + \Gamma(1 + \pi))-4 \pi \Gamma(1 + \pi))}{{\pi}^{\sqrt \pi} - \Gamma(1 + \pi)}}\color{red}{\approx55221,71}}\color{blue}{\neq2020}}$$

My questions are:

  • I wonder if there is a small typo in the formula?
  • Or is the formula far from accurate in any case?
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The outer $\pi$ is the prime-counting function.

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    $\begingroup$ Or, $\pi(17577.6)=2020$ $\endgroup$
    – QC_QAOA
    Jan 1 '20 at 21:13
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    $\begingroup$ In Mathematica, PrimePi[17577.61614280863]=2020 $\endgroup$
    – Pixel
    Jan 1 '20 at 21:13
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    $\begingroup$ @Henry so we have "two different pi" in the same formula? Do I understand correct? $\endgroup$
    – user548054
    Jan 1 '20 at 21:28
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    $\begingroup$ @Elementary in this case, yes, different meanings are intended. From a table of primes: 2018 17551 2019 17569 2020 17573 2021 17579 2022 17581 oeis.org/A000040/b000040.txt $\endgroup$
    – Will Jagy
    Jan 1 '20 at 21:40
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    $\begingroup$ From Wolfram MathWorld: "The notation pi(n) for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant pi=3.1415.... This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), "I am sorry about this; it is not my fault. You'll just have to put up with it." $\endgroup$
    – DJohnM
    Jan 1 '20 at 21:43

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