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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