Consider:
NPR1(n) =number of primes below 10
^{n}
NPR2(n)= number of primes with at most n digits
NPR3(n)= number of distinct prime divisors below (10
^{n})!
For a given n, which of the above is the biggest?
Rationalize your conclusion.
According to WolframAlpha, (10^3)! has 2877 prime factors, of which 168 are distinct.
168 is also the number of primes below 10^3, as well as the number of the number of primes with at most 3 digits.
It is true that there are numbers much smaller than (10^3)! with prime factors much larger than 10^3, but such numbers by definition will not appear in the factorial sequence until much later.
My impression is that these definitions are much the same.

Posted by broll
on 20170102 22:16:44 