Integers w/ 4 different partitions into 3 parts w/ the same product

Partition (number theory)
https://en.wikipedia.org/wiki/Partition_(number_theory)

Find integers   $N$   that have 4 different partitions into 3 parts with the same product

$N \; = \; a \; + \; b \; + \; c$
$N \; = \; d \; + \; e \; + \; f$
$N \; = \; g \; + \; h \; + \; i$
$N \; = \; j \; + \; k \; + \; l$

and with the same product,    $a \,b \,c \; = \; d \,e \,f \; = \; g \,h \,i \; = \; j \,k \,l$

math grad - Interest: Number theory
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4 Responses to Integers w/ 4 different partitions into 3 parts w/ the same product

1. pipo says:

I don’t know if it is the smallest solution but it fits:
(a, b, c) = (6, 56, 75)
(d, e, f) = (7, 40, 90)
((g, h, i) = (9, 28, 100)
(j, k, l) = (12, 20, 105)
So a + b +c = d + e + f = g + h + i = j + k + l = 137
And abc = def = ghi = jkl = 25200

pipo

2. paul says:

Here are the solutions where the total of a + b + c, etc is <=200. The smallest product is what Pipo found, though there are a few with a smaller sum.
Format is {sum, {{a,b,c},{d,e,f},g,h,i},{j,k,l}},product}

{118,{{21,25,72},{18,30,70},{15,40,63},{14,50,54}},37800}
{130,{{15,24,91},{14,26,90},{10,42,78},{9,56,65}},32760}
{133,{{24,25,84},{18,35,80},{16,42,75},{15,48,70}},50400}
{135,{{21,24,90},{15,36,84},{14,40,81},{12,60,63}},45360}
{137,{{12,20,105},{9,28,100},{7,40,90},{6,56,75}},25200}
{140,{{26,30,84},{21,39,80},{20,42,78},{18,52,70}},65520}
{149,{{30,39,80},{26,48,75},{25,52,72},{24,60,65}},93600}
{153,{{21,28,104},{16,39,98},{14,48,91},{13,56,84}},61152}
{155,{{9,26,120},{8,30,117},{6,45,104},{5,72,78}},28080}
{161,{{25,28,108},{20,36,105},{15,56,90},{14,72,75}},75600}
{167,{{16,25,126},{12,35,120},{10,45,112},{8,75,84}},50400}
{169,{{33,36,100},{30,40,99},{24,55,90},{22,72,75}},118800}
{174,{{21,25,128},{14,40,120},{12,50,112},{10,80,84}},67200}
{175,{{27,28,120},{18,45,112},{16,54,105},{14,80,81}},90720}
{182,{{14,18,150},{7,40,135},{6,50,126},{5,72,105}},37800}
{183,{{35,40,108},{30,48,105},{27,56,100},{24,75,84}},151200}
{189,{{34,35,120},{30,40,119},{21,68,100},{20,84,85}},142800}
{190,{{27,35,128},{24,40,126},{18,60,112},{16,84,90}},120960}
{194,{{18,24,152},{12,38,144},{9,57,128},{8,72,114}},65664}
{195,{{24,35,136},{16,60,119},{15,68,112},{14,85,96}},114240}

Paul.

3. pipo says:

And I think this is the smallest (both sum- and productwise) if we take 5 partitions:
Same format as Paul:
[185, (15, 44, 126), (12, 63, 110), (11, 84, 90), (18, 35, 132), (22, 28, 135), 83160)

pipo

4. paul says:

and for 6 partitions we have

{400,{{42,70,288},{36,84,280},{32,98,270},{28,120,252},{27,128,245},{24,180,196}},846720}
{427,{{52,60,315},{45,70,312},{36,91,300},{30,117,280},{27,140,260},{25,168,234}},982800}
{455,{{54,65,336},{40,91,324},{36,104,315},{35,108,312},{27,168,260},{26,189,240}},1179360}
{467,{{65,72,330},{54,88,325},{45,110,312},{40,130,297},{36,156,275},{33,200,234}},1544400}
{485,{{40,49,396},{33,60,392},{28,72,385},{16,154,315},{15,176,294},{14,231,240}},776160}
{491,{{45,56,390},{27,100,364},{26,105,360},{24,117,350},{20,156,315},{18,200,273}},982800}

Paul.