# Why are these numbers above the line?

The numbers above the line and below the line have a specific property to them:

\begin{array}{c} 1\ 3\ 5\ 6\ 7\ 9\ 10\ 11\ 12\ 13\ \ldots \\ \hline 2\ 4\ 8\ 15\ 16\ 21\ 30\ 32\ 33\ \ldots & \end{array}

Every natural number has a place either above or below the line.

What is this property? What are the next three numbers on each line? Would the numbers 29, 52, and 138 go above or below the line?

• No hits on OEIS. Once solved, if the property is of mathematical interest, it may be worth submitting. – nneonneo Feb 24 '15 at 21:43
• @nneonneo Are you sure? I get three results. – Doorknob Feb 24 '15 at 21:45
• The property is indeed a mathematical one, although I didn't think to check OEIS for it. – Joe Z. Feb 24 '15 at 21:45
• @Doorknob: the bottom sequence returns no hits. And none of the top three results describes my pattern. – Joe Z. Feb 24 '15 at 21:46
• I don't know if this is coincidence or not, but it appears that all (odd) primes are above the line, while (positive) powers of $2$ are below. – KSmarts Feb 24 '15 at 22:44

Numbers are below the line exactly if their number of distinct odd prime factors is even, except $1$ goes above the line. Equivalently, numbers are below the line when the sum of their distinct prime factors is even and positive.
So, $29$ would go above the line (odd prime), $52 = 2^2 \times 13$ would go above, and $138 = 2 \times 3 \times 23$ would go below.