## Carmichael Numbers

Carmichael numbers are numbers that pass Fermat's primality test, but they aren't actually prime numbers. They are Fermat Pseudoprimes.

See the sequence on OEIS

Numberphile covers Carmichael Numbers in this video... (starts at: 6:47)

Numberphile covers Carmichael Numbers in this video...

## Fibonacci Numbers

Fibonacci numbers are numbers that belong to the Fibonacci sequence which can be generated by repeatedly adding the previous two numbers of the sequence.

See the sequence on OEIS

Numberphile covers Fibonacci Numbers in this video...

## Fibonacci Sums

A Fibonacci sum (also called a Zeckendorf representation) is a way of representing any integer as a sum of non-consecutive Fibonacci numbers.

What we're doing here is: for every number, we're counting how many ways we can do this for that number (how many Zeckendorf representations it has) and coloring the circles darker the more ways we find.

If you select the Vogel Spiral (which is related to the Fibonacci sequence), you'll be able to see some interesting "boxy" patterns emerge.

See the sequence on OEIS

## Pythagorean Primes

Pythagorean Primes are prime numbers that can be written as the sum of two square integers.

It turns out that all Pythagorean Primes can be generated by the expression `4*n-1`

(but not for every `n`

). Try putting that in as a *custom* highlighting for comparison!

See the sequence on OEIS

Numberphile covers Pythagorean Primes in this video...

## Multiplicative Partition

A Multiplicative Partition is a way to write an integer as a product of other integers greater than 1.

For example, we can write `8`

as:

`8`

or `2×2×2`

or `2×4`

. So the number `8`

has `3`

Multiplicative Partitions.

We're coloring the numbers with more Multiplicative Partitions darker.

See the sequence on OEIS

## Vampire Numbers

Vampire Numbers are integers that can be factored into two numbers that have the same digits as the original.

For example, 1260 = 21 × 60, so 1260 is a Vampire Number, with *fangs*: 21, and 60

See the sequence on OEIS

Numberphile covers Vampire Numbers in this video...

## Random Numbers

A set of ordered pseudo-randomly generated integers.

Every number has a 50/50 chance of being highlighted. If you wrote this as a custom function it would be: `n * number(random() > 0.5)`

. Try it!

You can use this to compare a random pattern with other patterns.

We didn't use radioactive material to generate these random numbers... but Numberphile has done that...

## Random Opacity

We're highlighting each number by a random amount (between 0 and 1).

For every number we select a pseudo-random value between 0 and 1, and highlight the number by that value (1 means fully highlighted)

We didn't use radioactive material to generate these random numbers... but Numberphile has done that...

## Custom Highlighting

If you'd like, you can specify your own *custom* function to highlight the numbers. Just select the "custom" family and enter an equation in the "custom" text box. The input parameter "n" will count up from 1, and the numbers returned from the equation will be highlighted.

For example: `n^2 + 4*n + 1`

will highlight...

`6 (n=1), 13 (n=2), 22 (n=3), ... `