# partitions with the same product

Joseph Biberstine jrbibers at indiana.edu
Sun Jul 23 23:02:44 CEST 2006

```If I'm understanding you and we trust my code, here are the results on 4
to 100.  There are exactly three such partitions at these inputs: {39,
45, 49, 53, 62, 64, 65, 70, 71, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84,
85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100}, which is not
in OEIS and gets no hits from Superseeker.  No input under 100 has more
than four such partitions; it's not clear to me if you want exactly or
at least three such partitions.

This is the (hacky) Mathematica code:

pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
tanya[n_] := Max[Length /@ Split[Sort[pdt /@ Union[Partition[Last /@
Flatten[FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c},
Integers,(*failsafe*)PartitionsP[n]]], 3]]]]];
Select[Table[k,{k,4,100}], tanya[#]>=3 (*or strictly = ?*) &]

- JRB

Tanya Khovanova wrote:
> I am looking for a sequence of numbers that have 3 different partitions into 3 parts with the same product. I can't find it.
>
> I am interested in numbers that have two partitions too.
>
>
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```