Given an integer N, count how many ways it can be expressed as a product of M integers > 1.

Input is simply N and M, and output is the total count of distinct integer groups. Meaning you can use an integer more than once, but each group must be distinct (3 x 2 x 2 would not count if 2 x 2 x 3 is present).


1 < N < 231
1 < M < 30


Input 30 2 gives output 3, since it can be expressed 3 ways:

2 x 15
3 x 10
5 x 6

Input 16 3 gives output 1, since there's only one distinct group:

2 x 2 x 4

Input 2310 4 gives output 10:

5 x 6 x 7 x 11
3 x 7 x 10 x 11
3 x 5 x 11 x 14
3 x 5 x 7 x 22
2 x 7 x 11 x 15
2 x 5 x 11 x 21
2 x 5 x 7 x 33
2 x 3 x 11 x 35
2 x 3 x 7 x 55
2 x 3 x 5 x 77

Input 15 4 gives output 0, since it cannot be done.


Standard code golf loopholes apply, along with standard definitions for input/output. Answers may be a function or full program. Built-in functions for factorization and/or partitioning are not allowed, but others are fine. Code is counted in bytes.

  • \$\begingroup\$ What do you mean by partitioning ? \$\endgroup\$
    – Optimizer
    Jan 26, 2015 at 17:41
  • \$\begingroup\$ @Optimizer Grouping a list into nonoverlapping sublists. Some languages have this built in, such as Mathematica. \$\endgroup\$
    – Geobits
    Jan 26, 2015 at 17:43
  • \$\begingroup\$ Is there a time limit? A particularly naive algorithm could take centuries for a large value of M. Simple things like noting there can only be one factor larger than sqrt(N) obviously help a lot. \$\endgroup\$ Jan 26, 2015 at 17:51
  • 1
    \$\begingroup\$ @steveverrill Given the upper limit on N, there should only be 30 (prime) factors max, which speeds things up quite a bit. However, feel free to be naive. If your algorithm is not likely to provide an answer within a few hours, a well-explained proof of concept could help voters decide. \$\endgroup\$
    – Geobits
    Jan 26, 2015 at 17:54
  • \$\begingroup\$ a built in which allows you to do cartesian product of two list is allowed ? \$\endgroup\$
    – Optimizer
    Jan 26, 2015 at 18:21

6 Answers 6


Python 3, 59

f=lambda N,M,i=2:i<=N and f(N/i,M-1,i)+f(N,M,i+1)or-~M==N<2

We count up potential divisors i. With the additional argument i as the lowest allowed divisor, the core recursive relation is


For each i, we either choose to include it (possible as a repeat), in which case we divide the required product N by i and decrement M. If we don't, we increase i by 1, but only if i<N, since there's no use checking divisors greater than N.

When the minimum divisor i exceeds N, there's no more potential divisors. So, we check if we've succeeded by seeing if M==0 and N==1, or, equivalently, M+1==N==1 or M+1==N<2, since when M+1==N, the mutual value is guaranteed to be a positive integer (thanks to FryAmTheEggman for this optimization).

This code will cause a stack overflow for N about 1000 on most systems, but you can run it in Stackless Python to avoid this. Moreover, it is extremely slow because of its exponential recursive branching.

  • \$\begingroup\$ I think you can use -~M==N<2 \$\endgroup\$ Jan 26, 2015 at 22:15
  • \$\begingroup\$ @FryAmTheEggman I thought that would give false positives, but indeed it works, thanks to the joint constraints on M and N. Thanks! \$\endgroup\$
    – xnor
    Jan 26, 2015 at 22:25

Pyth - 24 23 22 21 bytes

Not a complicated solution. Will be golfing more. Just takes cartesian product of lists and filters. Same strategy as @optimizer (I'm guessing because of his comment, didn't actually decipher that CJam) Thanks to @FryAmTheEggman for 2 bytes and trick with M.


Defines a function g with args G and H

M                    function definition of g with args G and H
 l                   length of
  {                  set (eliminates duplicates)
   m                 map
    `Sd              repr of sorted factors so can run set (on bash escape ` as \`)
    f                filter
     q      G        equals first arg
      u*GHT1         reduce by multiplication
     ^     H         cartesian product by repeat second arg
       r2G           range 2 to first arg

Worked on all test args except last, was too slow on that one but there is no time limit given.

  • \$\begingroup\$ Input is fine in that format. \$\endgroup\$
    – Geobits
    Jan 26, 2015 at 20:11
  • 1
    \$\begingroup\$ Pyth golf tip: if you get 2 arguments, it is usually shorter to use M which defines the function g of 2 arguments, G and H. This is what I get for this: Ml{msdfqu*GHT1G^r2GH. Always nice to see another Pyth user :) \$\endgroup\$ Jan 26, 2015 at 22:37
  • \$\begingroup\$ @FryAmTheEggman updated thanks for the tip. \$\endgroup\$
    – Maltysen
    Jan 26, 2015 at 22:54
  • 1
    \$\begingroup\$ This seems to give an incorrect answer for the input 72 3, which returns 5, but there are in fact 6 answers, (2, 2, 18), (2, 3, 12), (2, 4, 9), (2, 6, 6), (3, 3, 8) \$\endgroup\$
    – isaacg
    Jan 27, 2015 at 9:25
  • 1
    \$\begingroup\$ @isaacg oh ok, I'll revert it back to my 21 char version. I didn't think summing it would work but it seemed to, so i'll go back to repr. Thanks for the catch. \$\endgroup\$
    – Maltysen
    Jan 27, 2015 at 20:32

Ruby, 67


Actually reasonably efficient for a recursive definition. For each divisor pair [d,q] of n, with d being the smaller one, we sum the result of f[q,m-1]. The tricky part is that in the inner calls, we limit factors to ones greater than or equal to d so that we don't end up double-counting.

1.9.3-p327 :002 > f[30,2]
 => 3 
1.9.3-p327 :003 > f[2310,4]
 => 10 
1.9.3-p327 :004 > f[15,4]
 => 0 
1.9.3-p327 :005 > f[9,2]
 => 1 

CJam, 48 bytes

This can be a lot shorter but I have added certain checks to make it work for decent number of M on the online compiler.


Try it online here

  • \$\begingroup\$ This is buggy. Try input 2 1. Expected output: 1. Actual output: 0. \$\endgroup\$ Jan 27, 2015 at 14:27
  • \$\begingroup\$ @PeterTaylor Sigh. Fixed. \$\endgroup\$
    – Optimizer
    Jan 27, 2015 at 19:45

T-SQL 456 373

I'm sure this'll break when the inputs are even close to being large.

Thanks to @MickyT for helping to save a lot of characters with CONCAT and SELECTing instead of multiple SETs.

SET @=1
SELECT @+=1,@C+=CONCAT(',# A',@),@D+=CONCAT('*A',@,'.A'),@E+=CONCAT(' AND A',@-1,'.A<=A',@,'.A')END
SET @C+=CONCAT(@D,'=',@N,@E)EXEC(@C)
  • \$\begingroup\$ I'd like to upvote this, but I can't find a simple way to test it. Any ideas? Third-party confirmation that it works is good, too. \$\endgroup\$
    – Geobits
    Jan 26, 2015 at 22:30
  • \$\begingroup\$ I get a couple of conversion errors we I run it (2012). They appear to be from these statements SET @C+=',# A'+@ and SET @D+='*A'+@+'.A'SET @E+=' AND A'+(@-1)+'.A<=A'+@+'.A' \$\endgroup\$
    – MickyT
    Jan 27, 2015 at 2:42
  • \$\begingroup\$ You'll also need to fix SET @C+=@D+'=@N'+@E+' SELECT @'. The @N is inside the quotes making it outside the scope when you execute @C. Also I think you will end up counting duplicates \$\endgroup\$
    – MickyT
    Jan 27, 2015 at 3:02
  • \$\begingroup\$ Now I've tested it on 2012. It should work. \$\endgroup\$
    – bmarks
    Jan 27, 2015 at 22:00
  • 2
    \$\begingroup\$ Works good now :) A couple of place where you can shave some characters. Try using CONCAT to build your strings. Then you can drop the CONVERTs. Try SELECT @+=1,@C+=CONCAT(...),@D+=CONCAT(...),@E+=CONCAT(...) in your WHILE loop. Should save you a reasonable number \$\endgroup\$
    – MickyT
    Jan 28, 2015 at 0:34

Jelly, 10 bytes


Try it online!

Assumes that "Built-in functions for factorization and/or partitioning are not allowed" doesn't prohibit the œċ builtin, which is combinations with replacements. If this isn't allowed, we can use a Cartesian product builtin, which is allowed, for 12 bytes: ḍƇ`ḊṗṢ€QP€ċ⁸

How it works

ḍƇ`ḊœċP€ċ⁸ - Main link. Takes n on the left and m on the right
 Ƈ         - Filter the integers, k, between 1 and n where the following is true:
ḍ          -   Does k divide...
  `        -   ...n?
           - This gives a list of divisors
   Ḋ       - Remove the first value, 1
    œċ     - Take all combinations of length m, with replacement
      P€   - Take the product of each
        ċ⁸ - Count occurrences of n

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