Cannon Ball Numbers

During production of our cannon ball number video, Matt supplied this extra information:

For the bigger numbers I checked until 10^11. Here is what I found (back in 2013)!
There are four triangle cannonball numbers below 10^9: 10, 120, 1540, 7140
There is one square cannonball number below 10^9: 4900
There is one pentagon cannonball number below 10^9: 946
There are two octagon cannonball numbers below 10^9: 1045, 5985
There is one decogon cannonball number below 10^9: 175
There is one 11-agon cannonball number below 10^9: 23725
There are two 14-agon cannonball numbers below 10^9: 441, 195661
There is one 17-agon cannonball number below 10^9: 975061
There is one 20-agon cannonball number below 10^9: 3578401
There is one 23-agon cannonball number below 10^9: 10680265
There is one 26-agon cannonball number below 10^9: 27453385
There is one 29-agon cannonball number below 10^9: 63016921
There is one 30-agon cannonball number below 10^9: 23001
There is one 32-agon cannonball number below 10^9: 132361021
There is one 35-agon cannonball number below 10^9: 258815701
There is one 38-agon cannonball number below 10^9: 477132085
There are two 41-agon cannonball numbers below 10^9: 55202400, 837244045
There is one 43-agon cannonball number below 10^9: 245905
There is one 50-agon cannonball number below 10^9: 314755
There is one 88-agon cannonball number below 10^9: 48280
There is one 145-agon cannonball number below 10^9: 101337426
There is one 276-agon cannonball number below 10^9: 801801
There is one 322-agon cannonball number below 10^9: 1169686
There is one 823-agon cannonball number below 10^11: 197427385
There is one 2378-agon cannonball number below 10^11: 432684460
There is one 2386-agon cannonball number below 10^11: 29437553530
There is one 31265-agon cannonball number below 10^11: 90525801730

And Matt sent this a day later (in April, 2019).

I just dusted off my old code and ran it unchanged on a much newer laptop and it was able (overnight) to check every value up to one trillion (10^12) for everything up to a megagon (one million sides).

It did not find anything bigger than a the 31265-agon but filled in a few gaps lower down (eg: there is a 44-agon cannonball number at 1,408,778,281). And to be fair, for the larger -agons checking to 10^12 is not enough as the numbers themselves are so big. I should re-code it slightly to raise the ceiling as the shapes get more sides.