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(3): some stats extended by Charlie)
It took about 24 hours but finally:
evens odds
906150257 453075129 453075128***
906150258 453075130 453075128***
906150259 453075131 453075128***
906150260 453075132 453075128***
906150261 453075132 453075129***
906150262 453075133 453075129***
906150263 453075134 453075129***
906150264 453075135 453075129***
906150265 453075135 453075130***
906150266 453075135 453075131***
...
so the answer is yes.

Posted by Charlie
on 20160111 22:06:41 