The last three digits of any countdown.
3-2-1
The smallest spenic Lucas number (322)
The Lucas numbers are closely related to the Fibonacci numbers: begin with L_0=2 and L_1=1 and for larger n, calculate L_n=L_{n-1}+L_{n-2}. That’s a different start seed, same iteration rule. Like the Fibonacci numbers, the ratio of successive terms approaches the golden ratio \phi, but the Lucas numbers approach L_n=\phi^n. L_{12}=322, and \phi^{12}\approx321.99689. On the other hand, the Fibonacci numbers approach \phi^n/\sqrt5; F_{12}=144, and \phi^{12}/\sqrt5\approx144.00139. Many Fibonacci relationships are easier to prove the analog for Lucas numbers, then use that for a Fibonacci proof. I actually referenced one of these proofs already in this topic: Unique counting - #161 by GeeTwo, which states that apart from zero and one, 144 is the only square Fibonacci prime.
Sphenic numbers are the product of three distinct primes, in this case 322=2*7*23.
322 is the 13^{th} Lucas number (including L_0=2), and the 35^{th} sphenic number.
I think thats the age when the Beatles could be handy mending a fuse when the lights go out, right?
The number of ways of drawing any number of nonintersecting chords joining 8 labeled points on a circle (323)
Chords are counted as intersecting if they share an endpoint or intersect inside the circle.
Number the 8 points clockwise (or counterclockwise) around the circle.
Zero chords: 1 case, no chords.
One chord: {8\choose2}=28 cases. Select 2 points, make a chord.
Two chords: 2{8\choose4}=2*70=140 cases. Select 4 points, {A B C D}. For each ABCD, AB,CD; AD,BC.
Three chords: 5{8\choose6}=5*28=140 cases. Select 6 points, {A B C D E F}. For each ABCDEF: AB,CD,EF; AB,CF,DE; AD,BC,EF; AF,BC,DE; AF,BE,CD.
Four chords: 14 cases: The seven cases where 1 connects to 2 or 4: 12,34,56,78; 12,38,45,67; 12,36,45,78; 12,38,45,67; 12,38,47,56; 14,23,56,78; 14,23,58,67; and the seven symmetric cases on reflection across the 1-5 axis where 1 connects to 6 or 8.
Finally, 1+28+140+140+14=323
Re-reading my post, I see that for 4 chords, I counted 12,38,45,67 twice and missed 12,34,58,67.
Also, not unique because team numbers have been used, but 323 is also Lights Out. My link is 2019, because I remember inspecting them at the first event I ever inspected, Rock City (now Arkansas) Regional in 2019. I’m not sure if I did their full inspection, but I clearly remember inspecting the ability of their software to limit extension outside the frame perimeter. We put the robot on blocks and placed a tall tool chest or something similar made of sheet metal the appropriate distance away. I directed the driver to “put the arm through its paces” or something similar. The loud clang a few seconds later when their manipulator slapped the sheet metal is why I remember it. Turns out it was a mis-calibrated sensor, and they passed a little bit later in the day.
The smallest number which is both triangular and the average of the squares of consecutive primes (325)
Smaller candidates
Here’s the list of triangular numbers up to 325:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325
Primes up through 19:
2, 3, 5, 7, 11, 13, 17, 19
Their squares:
4, 9, 25, 49, 121, 169, 289, 361
Average of two consecutives:
6½, 17, 37, 85, 140, 229, 325
There are no other such numbers less than aobout 5E13 (50 trillion).