10
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ No hits on OEIS. Once solved, if the property is of mathematical interest, it may be worth submitting. $\endgroup$ – nneonneo Feb 24 '15 at 21:43
  • 1
    $\begingroup$ @nneonneo Are you sure? I get three results. $\endgroup$ – Doorknob Feb 24 '15 at 21:45
  • $\begingroup$ The property is indeed a mathematical one, although I didn't think to check OEIS for it. $\endgroup$ – Joe Z. Feb 24 '15 at 21:45
  • $\begingroup$ @Doorknob: the bottom sequence returns no hits. And none of the top three results describes my pattern. $\endgroup$ – Joe Z. Feb 24 '15 at 21:46
  • 2
    $\begingroup$ 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. $\endgroup$ – KSmarts Feb 24 '15 at 22:44
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What about 2, 4, 8, and 16? They have no odd factors, but they are below the line. $\endgroup$ – Kevin Feb 25 '15 at 0:44
  • $\begingroup$ @Kevin You're right, I was trying to make an exception for 1 and got it wrong. Fixing. $\endgroup$ – xnor Feb 25 '15 at 1:03
  • $\begingroup$ Isn't this something to do with the Mobius mu function? $\endgroup$ – Rand al'Thor Feb 25 '15 at 1:26
  • $\begingroup$ You got it. The second one was what I was going for, and the exact formulation I had in mind was: "Numbers are above the line when the sum of their distinct prime factors is odd, with 1 counting as 1 by itself." $\endgroup$ – Joe Z. Feb 25 '15 at 6:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.