diameter #primtris with diam<=that ==================================== 2 0 4 0 8 3 16 7 ratio=2.333 32 26 ratio=3.714 64 78 ratio=3.000 128 266 ratio=3.410 256 752 ratio=2.827 512 2291 ratio=3.047 *** 1024 6332 ratio=2.764 2048 17415 ratio=2.750 4096 47313 ratio=2.717 8192 126061 ratio=2.664 16384 331121 ratio=2.627

From *** onward the ratio is decreasing... ultimately to 2+? That (and only that) limit would confirm my hypothesis that the asymptotic behavior is count=O(diam^(1+epsilon)). The "permanent ratio decrease" alone (without the limit) would prove that count=O(diam^1.4). Within the range of the table, though, the growth is more like count=0.16*diam^1.5.

[And the count for nonprimitive Herons should be at least a constant factor but at most a log factor greater than the count of primitive Herons, analogously to the way the count of integers is a log factor greater than the count of primes...]