a>b>cでa^2/(b+c)＋b^2/(c+a)＋c^2/(a+b)が自然数となる組について

\(a > b>c\)となる自然数の組のうち
\[\frac{a^2}{(b+c)}＋\frac{b^2}{(c+a)}＋\frac{c^2}{(a+b)}\]
が自然数となるようなものを考えたい。

現在見つかっている組
\((a,b,c)=\)
\((3k,2k,1k) \min k=5\)
\((5k,2k,1k) \min k=7\)
\((7k,3k,2k) \min k=5\)
\((5k,4k,3k) \min k=7\)
\((4k,2k,1k) \min k=10\)
\((5k,3k,1k) \min k=8\)
\((5k,4k,1k) \min k=9\)
\((11k,4k,1k) \min k=5\)
\((7k,5k,1k) \min k=8\)
\((7k,5k,3k) \min k=8\)
\((7k,5k,2k) \min k=9\)
\((33k,7k,2k) \min k=2\)
\((23k,7k,5k) \min k=3\)
\((5k,3k,2k) \min k=14\)
\((17k,3k,1k) \min k=5\)
\((17k,11k,4k) \min k=5\)
\((8k,3k,1k) \min k=11\)
\((8k,7k,3k) \min k=11\)
\((11k,5k,4k) \min k=9\)
\((3k,2k,1k) \min k=5\)
\((5k,2k,1k) \min k=7\)
\((7k,3k,2k) \min k=5\)
\((5k,4k,3k) \min k=7\)
\((4k,2k,1k) \min k=10\)
\((5k,3k,1k) \min k=8\)
\((5k,4k,1k) \min k=9\)
\((11k,4k,1k) \min k=5\)
\((7k,5k,1k) \min k=8\)
\((7k,5k,3k) \min k=8\)
\((7k,5k,2k) \min k=9\)
\((33k,7k,2k) \min k=2\)
\((23k,7k,5k) \min k=3\)
\((5k,3k,2k) \min k=14\)
\((17k,3k,1k) \min k=5\)
\((17k,11k,4k) \min k=5\)
\((8k,3k,1k) \min k=11\)
\((8k,7k,3k) \min k=11\)
\((11k,5k,4k) \min k=9\)
\((8k,7k,5k) \min k=13\)
\((11k,7k,3k) \min k=10\)
\((11k,9k,1k) \min k=10\)
\((16k,9k,5k) \min k=7\)
\((8k,6k,1k) \min k=14\)
\((14k,11k,10k) \min k=8\)
\((19k,11k,1k) \min k=6\)
\((23k,7k,1k) \min k=5\)
\((7k,2k,1k) \min k=18\)
\((9k,5k,1k) \min k=14\)
\((22k,5k,3k) \min k=6\)
\((6k,5k,4k) \min k=22\)
\((19k,16k,5k) \min k=7\)
\((17k,13k,7k) \min k=8\)
\((4k,3k,1k) \min k=35\)
\((9k,7k,5k) \min k=16\)
\((8k,7k,2k) \min k=18\)
\((29k,27k,13k) \min k=5\)
\((17k,7k,1k) \min k=9\)
\((11k,3k,1k) \min k=14\)
\((23k,5k,2k) \min k=7\)
\((33k,23k,7k) \min k=5\)
\((11k,5k,1k) \min k=16\)
\((13k,8k,2k) \min k=14\)
\((7k,6k,1k) \min k=26\)
\((183k,77k,13k) \min k=1\)
\((93k,7k,5k) \min k=2\)
\((17k,7k,4k) \min k=11\)
\((27k,13k,8k) \min k=7\)
\((9k,6k,1k) \min k=21\)
\((13k,8k,7k) \min k=15\)
\((41k,8k,7k) \min k=5\)
\((23k,13k,7k) \min k=9\)
\((13k,5k,3k) \min k=16\)
\((19k,5k,3k) \min k=11\)
\((19k,9k,2k) \min k=11\)
\((13k,12k,5k) \min k=17\)
\((23k,17k,7k) \min k=10\)
\((11k,10k,1k) \min k=21\)
\((13k,7k,5k) \min k=18\)
\((49k,17k,11k) \min k=5\)
\((129k,25k,11k) \min k=2\)
\((13k,3k,2k) \min k=20\)
\((65k,33k,7k) \min k=4\)
\((53k,13k,2k) \min k=5\)
\((19k,5k,2k) \min k=14\)
\((19k,11k,9k) \min k=14\)
\((9k,3k,1k) \min k=30\)
\((34k,21k,11k) \min k=8\)
\((17k,13k,3k) \min k=16\)
\((34k,29k,6k) \min k=8\)
\((21k,5k,4k) \min k=13\)
\((14k,6k,1k) \min k=20\)
\((7k,3k,1k) \min k=40\)
\((4k,3k,2k) \min k=70\)
\((15k,4k,1k) \min k=19\)
\((43k,37k,5k) \min k=7\)
\((19k,11k,5k) \min k=16\)
\((28k,27k,17k) \min k=11\)
\((14k,9k,1k) \min k=23\)
\((17k,3k,2k) \min k=19\)
\((25k,14k,1k) \min k=13\)
\((15k,13k,7k) \min k=22\)
\((6k,2k,1k) \min k=56\)
\((15k,8k,1k) \min k=23\)
\((15k,13k,8k) \min k=23\)
\((25k,11k,3k) \min k=14\)
\((25k,17k,3k) \min k=14\)
\((14k,11k,1k) \min k=25\)
\((71k,13k,7k) \min k=5\)
\((51k,29k,19k) \min k=7\)
\((8k,7k,1k) \min k=45\)
\((28k,11k,5k) \min k=13\)
\((23k,13k,3k) \min k=16\)
\((11k,6k,4k) \min k=34\)
\((6k,3k,1k) \min k=63\)
\((11k,4k,3k) \min k=35\)
\((35k,19k,9k) \min k=11\)
\((7k,4k,1k) \min k=55\)
\((7k,4k,3k) \min k=55\)
\((15k,11k,7k) \min k=26\)
\((13k,11k,2k) \min k=30\)
\((17k,15k,8k) \min k=23\)
\((9k,7k,2k) \min k=44\)
\((16k,11k,9k) \min k=25\)
\((31k,21k,5k) \min k=13\)
\((37k,7k,5k) \min k=11\)
\((18k,7k,5k) \min k=23\)
\((32k,17k,7k) \min k=13\)
\((19k,6k,5k) \min k=22\)
\((6k,4k,1k) \min k=70\)
\((25k,11k,9k) \min k=17\)
\((11k,2k,1k) \min k=39\)
\((11k,7k,2k) \min k=39\)
\((29k,11k,4k) \min k=15\)
\((19k,4k,1k) \min k=23\)
\((20k,19k,13k) \min k=22\)
\((37k,19k,5k) \min k=12\)
\((25k,19k,11k) \min k=18\)
\((41k,25k,19k) \min k=11\)
\((13k,2k,1k) \min k=35\)
\((35k,33k,17k) \min k=13\)
\((19k,5k,1k) \min k=24\)
\((12k,7k,2k) \min k=38\)
\((27k,8k,7k) \min k=17\)
\((17k,11k,10k) \min k=27\)
\((6k,5k,1k) \min k=77\)
\((11k,10k,4k) \min k=42\)
\((235k,17k,3k) \min k=2\)
\((19k,14k,11k) \min k=25\)
\((14k,4k,3k) \min k=34\)
\((9k,2k,1k) \min k=55\)
\((11k,7k,4k) \min k=45\)
\((31k,25k,17k) \min k=16\)
\((18k,17k,10k) \min k=28\)
\((46k,35k,9k) \min k=11\)
\((37k,15k,13k) \min k=14\)
\((10k,3k,2k) \min k=52\)
\((13k,7k,1k) \min k=40\)
\((13k,11k,9k) \min k=40\)
\((20k,19k,6k) \min k=26\)
\((25k,3k,2k) \min k=21\)
\((59k,31k,4k) \min k=9\)
\((77k,63k,58k) \min k=7\)
\((20k,7k,1k) \min k=27\)
\((17k,15k,1k) \min k=32\)
\((21k,19k,5k) \min k=26\)
\((43k,22k,17k) \min k=13\)
\((44k,21k,5k) \min k=13\)
\((41k,29k,1k) \min k=14\)
\((20k,9k,7k) \min k=29\)
\((83k,13k,8k) \min k=7\)
\((53k,37k,35k) \min k=11\)
\((9k,4k,1k) \min k=65\)
\((13k,7k,2k) \min k=45\)
\((31k,14k,5k) \min k=19\)
\((18k,15k,7k) \min k=33\)
\((17k,4k,3k) \min k=35\)
\((35k,33k,1k) \min k=17\)
\((35k,33k,16k) \min k=17\)
\((23k,9k,4k) \min k=26\)
\((86k,19k,9k) \min k=7\)
\((43k,22k,13k) \min k=14\)
\((55k,33k,17k) \min k=11\)
\((19k,17k,15k) \min k=32\)
\((12k,5k,3k) \min k=51\)
\((28k,5k,2k) \min k=22\)
\((77k,67k,3k) \min k=8\)
\((28k,27k,5k) \min k=22\)
\((39k,11k,5k) \min k=16\)
\((57k,20k,13k) \min k=11\)
\((19k,8k,3k) \min k=33\)
\((5k,4k,2k) \min k=126\)
\((29k,15k,7k) \min k=22\)
\((31k,11k,4k) \min k=21\)
\((41k,7k,1k) \min k=16\)
\((41k,15k,1k) \min k=16\)
\((17k,10k,3k) \min k=39\)
\((35k,3k,1k) \min k=19\)
\((35k,22k,3k) \min k=19\)
\((24k,11k,4k) \min k=28\)
\((21k,11k,1k) \min k=32\)
\((17k,7k,3k) \min k=40\)
\((20k,14k,1k) \min k=34\)
\((12k,7k,3k) \min k=57\)
\((38k,28k,17k) \min k=18\)
\((12k,9k,7k) \min k=57\)
\((43k,35k,13k) \min k=16\)
\((9k,5k,2k) \min k=77\)
\((7k,5k,4k) \min k=99\)
\((63k,47k,25k) \min k=11\)
\((24k,5k,1k) \min k=29\)
\((20k,15k,1k) \min k=35\)
\((9k,6k,4k) \min k=78\)
\((177k,112k,68k) \min k=4\)
\((21k,13k,5k) \min k=34\)
\((13k,9k,2k) \min k=55\)
\((11k,9k,2k) \min k=65\)
\((11k,9k,4k) \min k=65\)
\((55k,49k,1k) \min k=13\)
\((8k,2k,1k) \min k=90\)
\((20k,8k,1k) \min k=36\)
\((15k,9k,1k) \min k=48\)
\((9k,7k,1k) \min k=80\)
\((103k,23k,12k) \min k=7\)
\((7k,6k,2k) \min k=104\)
\((35k,21k,19k) \min k=21\)
\((44k,7k,5k) \min k=17\)
\((11k,6k,5k) \min k=68\)
\((47k,9k,7k) \min k=16\)
\((20k,18k,7k) \min k=38\)
\((51k,19k,9k) \min k=15\)
\((9k,8k,1k) \min k=85\)
\((9k,8k,7k) \min k=85\)
\((35k,9k,1k) \min k=22\)
\((43k,29k,13k) \min k=18\)
\((25k,24k,7k) \min k=31\)
\((37k,23k,5k) \min k=21\)
\((21k,19k,16k) \min k=37\)
\((46k,29k,5k) \min k=17\)
\((29k,20k,7k) \min k=27\)
\((11k,7k,1k) \min k=72\)
\((159k,28k,17k) \min k=5\)
\((14k,5k,1k) \min k=57\)
\((25k,7k,3k) \min k=32\)
\((23k,22k,13k) \min k=35\)
\((26k,9k,5k) \min k=31\)
\((101k,31k,9k) \min k=8\)
\((21k,5k,3k) \min k=39\)
\((13k,5k,2k) \min k=63\)
\((9k,5k,4k) \min k=91\)
\((13k,8k,1k) \min k=63\)
\((13k,8k,5k) \min k=63\)
\((25k,8k,7k) \min k=33\)
\((49k,19k,2k) \min k=17\)
\((15k,13k,11k) \min k=56\)
\((21k,19k,17k) \min k=40\)
\((47k,13k,5k) \min k=18\)
\((10k,7k,5k) \min k=85\)
\((15k,13k,6k) \min k=57\)
\((19k,17k,1k) \min k=45\)
\((27k,13k,5k) \min k=32\)
\((67k,11k,10k) \min k=13\)
\((46k,11k,9k) \min k=19\)
\((8k,3k,2k) \min k=110\)
\((11k,9k,7k) \min k=80\)
\((23k,16k,5k) \min k=39\)
\((18k,17k,7k) \min k=50\)
\((26k,9k,1k) \min k=35\)
\((26k,19k,9k) \min k=35\)
\((38k,25k,7k) \min k=24\)
\((17k,10k,7k) \min k=54\)
\((17k,10k,8k) \min k=54\)
\((22k,20k,13k) \min k=42\)
\((37k,13k,2k) \min k=25\)
\((24k,15k,1k) \min k=39\)
\((8k,5k,1k) \min k=117\)
\((27k,8k,1k) \min k=35\)
\((35k,19k,1k) \min k=27\)
\((73k,57k,47k) \min k=13\)
\((481k,329k,9k) \min k=2\)
\((23k,13k,5k) \min k=42\)
\((13k,12k,3k) \min k=75\)
\((14k,7k,1k) \min k=70\)
\((29k,5k,1k) \min k=34\)
\((76k,41k,15k) \min k=13\)
\((247k,185k,153k) \min k=4\)
\((43k,3k,2k) \min k=23\)
\((31k,9k,1k) \min k=32\)