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!