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number theory - Generating a list of all factorizations


What is the best way to generate a list of all factorizations of some number $n$? I'm quite new to Mathematica so this might be obvious. I have been trying some basic stuff with For-loops and FactorInteger and Divisors but I'm not really getting anywhere. There must be some elegant way of doing this. An example of the result I'm after, for $n=60$:


$$\{\{2,2,3,5\}, \{4,3,5\}, \{2,6,5\}, \{2,3,10\}, \{2,2,15\}, \{12,5\}, \{2,30\}, \{3,20\}, \{4,15\}, \{6,10\}, \{60\}\}.$$




Answer



A function from the article that cormullion linked is shorter and faster than what I proposed below. Transcribed in terse style:


uf[m_, 1] := {{}}
uf[1, n_] := {{}}
uf[m_, n_?PrimeQ] := If[m < n, {}, {{n}}]
uf[m_, n_] := uf[m, n] =
  Join @@ Table[Prepend[#, d] & /@ uf[d, n/d], {d, Select[Rest@Divisors@n, # <= m &]}]
uf[n_] := uf[n, n]

uf[60]



{{5, 3, 2, 2}, {5, 4, 3}, {6, 5, 2}, {10, 3, 2}, {10, 6}, {12, 5}, {15, 2, 2},
 {15, 4}, {20, 3}, {30, 2}, {60}}


I propose this:


ClearAll[f, f2, div]
mem : div[n_] := mem = Divisors@n
mem : div[n_, k_] := mem = # ~Take~ Ceiling[Length@#/k] &@div@n


f[n_, 1, ___] := {{n}}
mem : f[n_, k_, x_: 2] := mem =
  Join @@ Table[If[q < x, {}, {q, ##} & @@@ f[n/q, k - 1, q]], {q, n ~div~ k}]

f2[n_Integer] := Join @@ Table[f[n, i], {i, Tr@FactorInteger[n][[All, 2]]}]

The function f2 finds all factorizations as requested:


f2[60]



 {{60}, {2, 30}, {3, 20}, {4, 15}, {5, 12}, {6, 10}, {2, 2, 15}, {2, 3, 10},
    {2, 5, 6}, {3, 4, 5}, {2, 2, 3, 5}}

It is quite fast:


f2[1080^2] // Length // Timing


{0.109, 16434}


The function f (upon which f2 is written) efficiently finds the factorizations of n of length k with a minimum factor of x:


f[60, 2, 5]


{{5, 12}, {6, 10}}

It is optimized with memoization as described here. It can be written without memoization (as shown there) to use less memory but computation will take longer.


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