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?

OEIS A008836: 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, ...

"Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity)."

Basically, 1 = n is primeeven and -1 = n is primeodd.

OEIS A02819: 0, 1, 0, -1, 0, -1, 0, -1, -2, -1, 0, ...

"Liouville's function L(n) = partial sums of A008836."

This is positive when primeevens are in the lead, negative when primeodds are in the lead, and 0 when tied.

In the comments:

"George Polya conjectured in 1919 that L(n) <= 0 for all n >= 2. .... In 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257."

Charlie successfully verified this result. Congratulations.