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3 equal products (Posted on 2018-11-21) Difficulty: 3 of 5
Lets take a number 12 as an example . It can be partitoned into three integer summands in 12 ways:
12=1+1+10
12=1+2+9
12=1+3+8
12=1+4+7
12=1+5+6
12=2+2+8
12=2+3+7
12=2+4+6
12=2+5+5
12=3+3+6
12=3+4+5
12=4+4+4

Multiplying the 3 members of each partitions results in 12 distinct numbers: 10,18,...60,64.

On the other hand the same treatment applied to number 13 produces a pair of equal results: 13=1+6+6=2+2+9 and 1*6*6=2*2*9=36 (a well known problem of children's ages).

Find the smallest number which has 3 distinct partitions into 3 parts, each of them with the same product.

Bonus: list all numbers below 1000 boasting this feature.

No Solution Yet Submitted by Ady TZIDON    
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re(2): Answer to the bonus. | Comment 9 of 10 |
(In reply to re: Answer to the bonus. by Ady TZIDON)

My comment (edited) is now in the A119028 entry in OEIS. The second clause was added by the reviewer whose name appears also. Apparently they feel it's not just a conjecture, but I have no idea of a proof.
  Posted by Charlie on 2018-11-24 09:18:48

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