# Is this $2020$ holiday formula correct? $\pi\left( \dfrac{\left( \pi!\right)!-\lceil \pi \rceil \pi! }{{\pi}^{\sqrt \pi}-\pi!}\right)=2020$

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?

The outer $$\pi$$ is the prime-counting function.

• Or, $\pi(17577.6)=2020$ – QC_QAOA Jan 1 '20 at 21:13
• In Mathematica, PrimePi[17577.61614280863]=2020 – Pixel Jan 1 '20 at 21:13
• @Henry so we have "two different pi" in the same formula? Do I understand correct? – Elementary Jan 1 '20 at 21:28
• @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 – Will Jagy Jan 1 '20 at 21:40
• 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." – DJohnM Jan 1 '20 at 21:43