User:EricABQ/Pandemic googologism

This is a googologism that grows with time, similar to the Lynz. I first defined it on July 12, 2021. It was inspired by the seemingly endless lockdowns and lifestyle changes caused by the COVID pandemic, and the ever-growing number of cases.

Definition
\( P(n) = 10^{H_\beta(6)} \), where:
 * \( \beta = \max \{\alpha < \psi_0(\varepsilon_{\Omega_\omega + 1}) | g_\alpha(6) = n(n + 1)/2\}) \)
 * \( g_\alpha(n) \) is the slow-growing hierarchy using Wainer hierarchy fundamental sequences when \( \alpha < \varepsilon_0 \), then fundamental sequences for Buchholz's function when \( \varepsilon_0 \le \alpha < \psi_0(\varepsilon_{\Omega_\omega + 1}) \).
 * \( H_\alpha(n) \) is the Hardy hierarchy using the same fundamental sequences

To find the current value of \( P \), the pandemic googologism, find \( P(t) \) where \( t \) is the number of days since March 11, 2020 (the number updates at 12 AM Mountain time). I chose March 11, 2020 because that is the day the coronavirus first reached the town my college was in and my life got changed forever (even though it took a few days before I started quarantining). It was also the day COVID-19 was officially declared a pandemic.

Example values
Here are the values for the first few days \( P \):

10^6 --- March 11, 2020

10^7

10^9

10^12 --- March 14, 2020: I start quarantining

10^20

10^36

10^72

10^160

10^384

10^4718592

10^4.04e18 (this and all the following values are approximate)

10^1.21e60

10^10^14801

10^10^1.21e18

10^10^4.31e88

10^10^10^121210694

10^10^10^7.04e49

10^10^10^10^1420443

50th day: 10^^10^^10^^10^^10^^10^10^10^10^744718456231

100th day: 10^^^10^^^10^^^10^^10^^10^^10^^10^^10^6313064

305th day: 10{17}17|18 (\(f_\omega(18)\) to be exact; this is the first time \( P \) exceeds \( f_\omega(6) \))

365th day: (1,0)|10{4}10{4}10^^^10^^^10^^^10^^10^^10^^2.47e12 (using PsiCubed's array notation)

Value on July 12, 2021 (when I originally came up with this definition): (1,0)|(1,0)|10{4}10{4}10{4}10^^^10^^^10^121210694

Value on March 20, 2022: (1,0)|(1,0)|(1,0)|(1,0)|(1,0)|10{4}10{4}10{4}10{4}10{4}10^^10^^10^^10^^10^^10^10^10^10^10^6313063