# TotallyOriginallyCreatedByMe Conjecture

Computer Science Level pending

Consider segregating positive integers into two categories:

• Even-type: if its prime factorization has an even number of prime(s), or it equals to $$1$$.

• Odd-type: if its prime factorization has an odd number of prime(s).

Define

• $$E(N)$$ to be the number of positive integers of even-type that are less than or equals to $$N$$

• $$O(N)$$ to be the number of positive integers of odd-type that are less than or equals to $$N$$

I claim that $$E(N) \leq O(N)$$ for all positive integers $$N$$ greater than $$1$$ but less than $$10^9$$. Am I right?

Details and assumptions:

• $$18 = 2 \times 3 \times 3$$, which has $$3$$ prime factors, so $$18$$ is an odd-type

• $$19 = 19$$, which has $$1$$ prime factor, so $$19$$ is an odd-type

• $$33 = 3 \times 11$$, which has $$2$$ prime factors, so $$33$$ is an even-type

• If $$N=25$$, then $$1,4,6,9,10,14,15,16,21,22,24,25$$ are even-type, $$2,3,5,7,8,11,12,13,17,18,19,20,23$$ are odd-type. So $$E(25)= 12, O(25)=13$$. Which means $$E(25) \leq O(25)$$, thus it's true for $$N=25$$.

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