Triples of positive integers with the same sum and the same product

How many distinct triples (a,b,c) of positive integers can you have with the same sum and the same product ?
"Unsolved Problems in Number Theory" by Richard Guy - Section D16 - p. 96 go.
Ex: the 3 triples (4,15,20) (5,10,24) (6,8,25) have ∑=39, and ∏=1200.

Previous results

With 2 triples, inifinite solutions, smallest :

∑=19 for (2,8,9), (3,4,12)
∏=90 for (1,9,10), (2,3,15)

With 3 triples, inifinite solutions, smallest :

∑=39 for (4,15,20), (5,10,24), (6,8,25)
∏=1200 for (4,15,20), (5,10,24), (6,8,25)

With 4 triples, inifinite solutions, smallest :

∑=118 for (14,50,54), (15,40,63), (18,30,70), (21,25,72)J.G. Mauldon
∏=25200 for (6,56,75), (7,40,90), (9,28,100), (12,20,105)J.G. Mauldon

With 5 triples, single solution :

∑=981 for (6,480,495), (11,160,810), (12,144,825), (20,81,880), (33,48,900)J.G. Mauldon
∏=1425600 for (6,480,495), (11,160,810), (12,144,825), (20,81,880), (33,48,900)J.G. Mauldon

Contributions (last update : 2025 May 6)

With 5 triples, several solutions found:

triples	∑	∏	a_min	a_max
(11,84,90) (12,63,110) (15,44,126) (18,35,132) (22,28,135)	185	83160	11	135
(8,168,270) (10,112,324) (14,72,360) (20,48,378) (27,35,384)	446	362880	8	384
(6,253,504) (8,161,594) (9,138,616) (11,108,644) (24,46,693)	763	765072	6	693
(8,132,315) (9,110,336) (11,84,360) (16,54,385) (24,35,396)	455	332640	8	396
(5,792,855) (9,275,1368) (11,216,1425) (15,152,1485) (25,88,1539)	1652	3385800	5	1539
(5,396,1680) (6,315,1760) (8,225,1848) (12,144,1925) (21,80,1980)	2081	3326400	5	1980
(4,1666,2430) (5,1071,3024) (14,306,3780) (27,153,3920) (51,80,3969)	4100	16193520	4	3969
(4,1188,3451) (6,693,3944) (11,348,4284) (24,153,4466) (29,126,4488)	4643	16399152	4	4488
(3,3472,4554) (4,1953,6072) (22,279,7728) (33,184,7812) (42,144,7843)	8029	47434464	3	7843
...
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With 6 triples, several solutions found:

triples	∑	∏	a_min	a_max
(24,180,196) (27,128,245) (28,120,252) (32,98,270) (36,84,280) (42,70,288)	400	846720	24	288
(14,231,240) (15,176,294) (16,154,315) (28,72,385) (33,60,392) (40,49,396)	485	776160	14	396
(12,728,765) (13,540,952) (21,260,1224) (24,221,1260) (35,144,1326) (54,91,1360)	1505	6683040	12	1360
(11,1024,1155) (12,770,1408) (16,480,1694) (22,320,1848) (28,242,1920) (30,224,1936)	2190	13009920	11	1936
...
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With 7 triples, several solutions found:

triples	∑	∏	a_min	a_max
(35,216,260) (36,195,280) (40,156,315) (42,144,325) (45,130,336) (60,91,360) (72,75,364)	511	1965600	35	364
(33,252,260) (35,198,312) (40,154,351) (42,143,360) (52,108,385) (56,99,390) (65,84,396)	545	2162160	33	396
(23,416,450) (24,345,520) (25,312,552) (26,288,575) (32,207,650) (39,160,690) (75,78,736)	889	4305600	23	736
(12,728,765) (13,540,952) (21,260,1224) (24,221,1260) (35,144,1326) (54,91,1360) (68,72,1365)	1505	6683040	12	1365

With 8 triples, several solutions found:

triples	∑	∏	a_min	a_max
(70,432,520) (72,390,560) (80,312,630) (84,288,650) (90,260,672) (91,256,675) (120,182,720) (144,150,728)	1022	15724800	70	728
(51,880,1092) (52,816,1155) (56,680,1287) (72,455,1496) (77,416,1530) (80,396,1547) (105,286,1632) (156,187,1680)	2023	49008960	51	1680
(42,1680,2288) (44,1470,2496) (48,1232,2730) (52,1078,2880) (98,480,3432) (126,364,3520) (130,352,3528) (195,231,3584)	4010	161441280	42	3584

Note: 🏆∑₈ = 2 * 🏆∑₇.

With 9 triples, several solutions found:

triples	∑	∏	a_min	a_max
(99,588,600) (100,539,648) (105,462,720) (112,405,770) (126,336,825) (132,315,840) (162,245,880) (165,240,882) (196,200,891)	1287	34927200	99	891

view numbers as 3600000, 3,600,000, 3.600.000.

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