## Peloton Numbers

### What is a Peloton Number?

The concept of peloton numbers was introduced in the September 22, 2023 column of Fiddler on the Proof, by Zach Wissner-Gross.

A peloton formation can have any shape between a triangle and a rhombus. For example, here are the possible formations for a peloton of width 4:

A peloton number is the number of dots in a peloton formation. All trianglar numbers are peloton numbers. All perfect squares are peloton numbers (the largest peloton possible of a given width). And there are additional peloton numbers corresponding to the formations between these extremes.

### Peloton Numbers up to 150

There are 78 peloton numbers between 1 and 150. Thirteen of them are double peloton numbers (that have two different formations each). And one is even a triple peloton number (that has three different formations).

The first seven double peloton numbers are illustrated below:

And here is the first triple peloton number.

### Higher Multiplicities

It is possible to have quadruple peloton numbers and even higher multiplicities. The table below lists the smallest peloton numbers that have exactly n formations.

Some of the entries show — ? — after their multiplicity. That means I haven’t found a suitable peloton number to go in that position. I have checked all numbers up to 64,000,000,000, so the missing numbers must be greater than that (if they even exist).

 n peloton number 1 1 2 15 3 141 4 610 5 6,903 6 2,395 7 338,241 8 10,606 9 40,713 10 117,349 11 812,116,341 12 98,190 13 39,793,700,703 14 5,750,095 15 1,994,931 16 434,841 17 — ? — 18 1,669,228 19 — ? — 20 4,811,304 21 97,751,613 22 13,805,977,795 23 — ? — 24 3,489,333 25 576,535,023 26 — ? — 27 29,028,280 28 235,753,890 29 — ? — 30 81,792,166 31 — ? — 32 21,307,203

 n peloton number 33 — ? — 34 — ? — 35 28,250,216,121 36 59,318,659 37 — ? — 38 — ? — 39 — ? — 40 170,977,311 41 — ? — 42 4,007,816,128 43 — ? — 44 — ? — 45 1,422,385,714 46 — ? — 47 — ? — 48 143,062,648 49 — ? — 50 23,637,935,938 51 — ? — 52 — ? — 53 — ? — 54 1,190,159,475 55 — ? — 56 8,377,888,233 57 — ? — 58 — ? — 59 — ? — 60 2,906,614,285 61 — ? — 62 — ? — 63 — ? — 64 1,451,064,000

 n peloton number 65 — ? — 66 — ? — 67 — ? — 68 — ? — 69 — ? — 70 — ? — 71 — ? — 72 2,432,065,014 73 — ? — 74 — ? — 75 — ? — 76 — ? — 77 — ? — 78 — ? — 79 — ? — 80 7,010,069,746 81 42,294,203,778 82 — ? — 83 — ? — 84 — ? — 85 — ? — 86 — ? — 87 — ? — 88 — ? — 89 — ? — 90 58,317,814,269 91 — ? — 92 — ? — 93 — ? — 94 — ? — 95 — ? — 96 10,443,573,295

### Multiplicities in Order by Peloton Number

The following chart gives the same information as the previous chart but this time in order by peloton number.

The shaded rows are peloton numbers that have a greater multiplicity that any smaller peloton number. Those multiplcities are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 80, 96, ....

What might the next number in this sequence be? I'm guessing 108, then 120 or 128.

### Density of Peloton Numbers

How likely is it that a given random integer is a peloton number? For numbers less than 10,000, it is over 40%. But the probability decreases as numbers get larger.

For numbers greater than 100 million, the density of peloton numbers is less than 30%. The following chart shows the trend.

On the other hand, the average multiplicity of peloton numbers increases as the peloton numbers get larger.

These values have an inverse relationship:

Even though the density and multiplicity of peloton numbers change as the numbers get larger, C stays constant. And it turns out that C is a specific transcendental number:

(I discovered this imperically. I wonder if there is a way to derive this mathematically?)

We could call C the formation density; it is the ratio of the number of peloton formations to the number of dots available to be used.

For example, there are 623,244 peloton formations containing one million dots or less. The ratio of the number of formations to the number of available dots is 0.623244.

### Peloton Spiral

With a linear series of numbers like peloton numbers, a fun way to visualize them is arrange them in a square spiral, like this:

All peloton numbers are shaded in red; the double peloton numbers are shaded in darker red.

Zooming out you can discern some patterns:

There is a dominant diagonal line running from northwest to southeast. These are perfect squares, since every perfect square is also a peloton number.

There are fainter lines running parallel to these, at offsets of 1, 3, 5, 7, etc. These correspond to the numbers you get by lopping off the bottom row (1 dot) of a “full” peloton, then lopping off the next row (2 dots), etc.