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
some stats by Charlie)
Extended to 100,000,000; none found other than 1:
1 1 0***
10000000 4999579 5000421
20000000 9997745 10002255
30000000 14997185 15002815
40000000 19998404 20001596
50000000 24996196 25003804
60000000 29998492 30001508
70000000 34996701 35003299
80000000 39996434 40003566
90000000 44998912 45001088
100000000 49998058 50001942

Posted by Charlie
on 20160110 17:56:21 