Number · OEIS A002858

The Arc They Never Land On

Start with 1 and 2. Forever after, write down the smallest whole number that is a sum of two different earlier numbers in exactly one way. That rule is the whole of it, and it hides nothing on purpose. Yet the numbers it makes keep a beat: there is a frequency, near 2.5714, at which they all fall into step. Lay them around a circle tuned to that beat and they crowd into two arcs and leave a third arc empty, forever, with only four small exceptions below a million. This page builds the sequence in front of you, lets you hunt down the frequency yourself, and shows the empty arc open up. What it cannot do is tell you why the arc is there. Nobody can.

The sequence is Stanislaw Ulam's, from 1964, and for a while it was just a curiosity: a sequence defined by a one-line rule whose large-scale shape resisted every attempt to describe it. Then in 2015 Stefan Steinerberger noticed something that should not be there.¹ These numbers, born of nothing but addition, behave as if they were tuned to a hidden radio station. This page is that station, received live.

The rule, and only the rule

Here is the machine. A number joins the sequence when it can be written as a + b with a < b, both already in the sequence, in one and only one way. Zero ways: skipped. Two or more ways: skipped. Watch it tick.

Nothing in that rule mentions circles, or frequencies, or the number 2.5714. It is pure bookkeeping about sums. Hold on to that, because everything below is going to feel like it was smuggled in from another problem.

Lay them on a circle

Take a circle. Give it a circumference of L. Now walk each Ulam number around it: the number a lands at the point you reach after walking a distance a, wrapping around as many times as needed. This is just a reduced modulo L, drawn as an angle. For almost every choice of L the landing points smear evenly around the ring, the way you would expect a sequence of numbers with nothing to hide to behave.

Then you reach L ≈ 2.443. Drag the dial to it.

What you are looking at, said in the flat language of trigonometry: put α = 2π/L. Then almost every Ulam number a satisfies cos(α·a) < 0. The cosine is negative exactly on the far arc of the circle, the arc away from the starting point, which is why the near arc stays empty. That is the whole of Steinerberger's observation. The mystery is not in the statement; it is in the fact that it is true.

Find the frequency yourself

You should not have to take 2.443 on faith, so do not. Here is a way to hear the sequence for a frequency without knowing it in advance. For a trial frequency α, add up the little unit arrows e^iαa, one per Ulam number, and measure how long the total is. If the numbers are smeared evenly the arrows cancel and the total is nearly zero. If they march in step at that frequency, the arrows align and the total is long. Sweep α and watch for a spike.

The spike sits at α ≈ 2.5714. Press Zoom in and refine and the browser will chase the value against longer and longer stretches of the sequence, and the refined number, found here from the sequence alone with no constant put in by hand, agrees with Steinerberger's published 2.5714474995 to six decimal places on the modest piece this page holds in memory. The beat is not an artifact of how many terms you look at: as the sequence grows the spike does not fade the way a random coincidence would. It holds its height. That steadiness is the difference between a signal and a fluke.

The count, and the four that disobey

With the frequency in hand, the claim becomes something you can tally exactly. Of the first 14,975 Ulam numbers, every one below 200,000, how many actually land on the far arc, the negative side?

What is true, and what is only observed

The line between the two is the whole point of this place, so here it is, drawn sharply.

True, and checked in your browser just now. The sequence above is the Ulam sequence, generated two independent ways that agree on every term tested (the first 1,500). The frequency α was found by searching, not assumed, and it matches the published value. Reduced to that frequency, 99.97 percent of the 14,975 terms below 200,000 fall on the negative arc, with the mean of cos(α·a) sitting near -0.79 rather than the 0 a shapeless sequence would give. The only exceptions are 2, 3, 47 and 69. The offline verifier runs the same tally over all 74,084 terms below a million: 99.99 percent on the negative arc, and the same four exceptions.

Observed, but not proven, by anyone. Everything about why. There is no proof that the signal persists forever. There is no proof that the list of four exceptions never grows. The constant 2.5714474995 is not known to equal any simpler expression, or to connect to any other constant in mathematics; it appears to belong to this sequence and no other. Steinerberger's paper is titled, with due care, a report of a hidden signal, not an explanation of one. A rule a child could follow produces an order that the whole field cannot yet account for, and that gap between the simplicity of the cause and the strangeness of the effect is not a flaw in the telling. It is the finding.