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I saw this a couple of days ago on ueno's brand redesign and have been trying to figure out the pattern between the colors for the past few days. Can anyone offer some insight. Been working on my color theory but can't seem to find the pattern and therefore the odd one out. I thought it could be some math around the hue values or some consistency in the saturation brightness but can't see any patterns jumping out. I then tried testing to see if the perceived brightness was identical across the colors which it isn't.

You can see the colors below:

Fancy and bright as described in the text is not enough pattern?

This is the color wheel:

This is the same color wheel with a hight brightness effect:

If I had to describe the colors I would say pure hues with high bright value and black

00CC76 is redundant and a bit too dark for the scheme. There aren't two of any other color aside from green.

Edit: Added a little more support to my statement and an image:

I think it's safe to say that black belongs in the group to be used as as the highest contrast, and thus "brightest" way to display text against a white background. This color palette is obviously intended to be used over a white background. So, discounting black, every hue but one has a comfortable home in the set, either a primary or secondary color. Of the two greens, the lighter fits the bright scheme best. All of the other secondary colors are somewhat softer. 00CC76 is more of a true green, so it doesn't belong in this palette.

... can't seem to find the pattern and therefore the odd one out.

It's #3EC2F1.

Source: The online Image Color Summarizer provided this answer:

Color Clusters

Colors in the image were clustered into 5 groups (k-means). The average color of the colors for each cluster is shown. The name is the closest named color and its distance is shown using ΔE. The tags are the set of words formed by all named neighbours within ΔE ≤ 5. The list of words above is the set of all unique words in this set of words.

Notice that one color stands alone, black is not included as a color.

There are also available a set of histograms for various colorspaces which I did not include. My results can be repeated by screenshotting the image that you provided and uploading it to the website.