Recently, the new largest known prime, 2^136,279,841-1 was discovered. This prime has 41 million digits, surpassing the previous prime that has 24 million digits.
Let's do some derivatives of this number, shall we?
But... there's also other primes as well. All of these numbers are dynamic googolisms.
Largest known prime = 2^136,279,841-1
Largest known perfect number = 2^136,279,840(2^136,279,841-1)
Largest known semiprime = (2^136,279,841-1)^2
Largest known squarefree semiprime = (2^136,279,841-1)(2^82,589,933-1)
Largest known sphenic number = (2^136,279,841-1)(2^82,589,933-1)(2^77,232,917-1)
Largest known repunit prime = (10^86453-1)/9
Largest known repunit semiprime = ((10^86453-1)/9)^2
Largest known squarefree repunit semiprime = ((10^86453-1)/9)*((10^49081-1)/9)
Largest known generalized Fermat prime = 4*5^11786358+1
Largest known generalized Fermat semiprime = (4*5^11786358+1)^2
Largest known emirp = 10^10,006 + 941992101 * 10^4999 +1
Reversed largest known emirp = 10^10,006 + 101299149 * 10^4999 +1(**)
Largest known Proth prime = 10223*2^31172165+1
Largest known Sierpinski prime (second, k prime) = 168451*2^19375200+1
Largest known Sierpinski prime (second, k composite) = 202705*2^21320516+1