Let me pose a problem here. I am not sure if it is there on Project Euler, but I am sure Project Euler has many problems which have similar taste.
Consider the function , where
is the number of primes
. Denote
,
, and so on. This gives a strictly decreasing sequence, and hence
for some
. This
depends on
. I would take a liberty to call this
the
-length of
and the sequence
, the
-sequence of
?
Now I can ask several questions.
- Find a prime number whose
-sequence consists only of prime numbers? Larger the better!
- Find a natural number whose
-sequence contains no prime other than 2. Again, larger the better.
- Among the natural numbers
, which one has largest
-length?
- Plot the function
-length.
I am sure you may also ask plenty of questions based on this.
Enjoy Pythoning!