Let a primeven be a positive integer that is the product of an even number of primes. Let a primeodd be a positive integer that is the product of an odd number of primes. Then, 1 is a primeven because it is the product of 0 primes. 2 is a primeodd because it is the product of 1 prime. 3 is a primeodd because it is the product of 1 prime. 4 is a primeven because it is the product of 2 primes. Here are the first 10 positive integers.
Number: Factorization: Number of primes: Type:
1 0 primeven
2 2 1 primeodd
3 3 1 primeodd
4 2*2 2 primeven
5 5 1 primeodd
6 2*3 2 primeven
7 7 1 primeodd
8 2*2*2 3 primeodd
9 3*3 2 primeven
10 2*5 2 primeven
Suppose the primevens and primeodds had a race. First, the primevens would be ahead because 1 is a primeven. Then, there would be a tie because 2 is a primeodd. Then, the primeodds would be ahead because 3 is a primeodd. Then, there would be a tie because 4 is a primeven. Here are the winners from 1 to 10.
Number: Type: Primevens: Primeodds: Winner:
1 primeven 1 0 primevens
2 primeodd 1 1 tie
3 primeodd 1 2 primeodds
4 primeven 2 2 tie
5 primeodd 2 3 primeodds
6 primeven 3 3 tie
7 primeodd 3 4 primeodds
8 primeodd 3 5 primeodds
9 primeven 4 5 primeodds
10 primeven 5 5 tie
The primevens were ahead at the start, but have not been ahead since then. Do the primevens ever become the winner again?
(In reply to
re(2): some stats extended by Jer)
I've had the program working overnight, and what with the increasing difficulty of factoring larger numbers, it's only up to just over 520,000,000, and probably slowing down as the numbers get bigger.

Posted by Charlie
on 20160111 07:39:15 