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?

Every prime is primeodd. A000040 in Sloane.

Some composite numbers are composite with an odd number of prime factors with multiplicity, or COPs for short.

If a number is neither a prime nor a COP, then it is primeven.

Sloane A046339 has a list of COPS, to which the primes must be added:

≤    P    COP    Total    primeven
10   4       1       5            5
50  15    13      28          22
100 25   26      51          49
150 35   42      77          73

Primes plus COPS are thus consistently more than primevens, for small values.

Sloane's list of COPS is a bit short, but there is reason to think that this pattern will change. As the numbers get larger, the primes thin out a bit and so contribute less to the total, whereas the number of primevens is already significantly larger than the number of primeodds.

Now we have the two series of numbers 1,4,6,9,10... and 2,3,5,7,8,11..., primeevens and primeodds. But as soon as an additional prime is included, say, 5*1, 5*2*2, 5*2*3,...5*2,5*3, 5*5... etc, then the primevens become primeodds and vice versa. Call these numbers P, 2P,...50P etc; primevens seem to predominate quite strongly; 29 versus 21 up to 50P, for example. So the fact that primes + COPS dominate to start with is itself at least suggestive that at some point this situation must reverse.

Edited on January 11, 2016, 11:16 pm

Posted by broll on 2016-01-10 12:30:36