Number of sums of factors

Question

Given a positive integer n > 1 determine how many numbers can be made by adding integers greater than 1 whose product is n. For example if n = 24 we can express n as a product in the following ways

24 = 24             -> 24            = 24
24 = 12 * 2         -> 12 + 2        = 14
24 = 6 * 2 * 2      -> 6 + 2 + 2     = 10
24 = 6 * 4          -> 6 + 4         = 10
24 = 3 * 2 * 2 * 2  -> 3 + 2 + 2 + 2 = 9
24 = 3 * 4 * 2      -> 3 + 4 + 2     = 9
24 = 3 * 8          -> 3 + 8         = 11

We can get the following numbers this way:

24, 14, 11, 10, 9

That is a total of 5 numbers, so our result is 5.

Task

Write a program or function that takes n as input and returns the number of results that can be obtained this way.

This is a code-golf question so answers will be scored in bytes, with fewer bytes being better.

OEIS sequence

OEIS A069016

Answer 1

Brachylog, 8 bytes

{~×≜+}ᶜ¹

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Explanation

{    }ᶜ¹  Count unique results of this predicate:
 ~×       Create list of numbers whose product is the input.
   ≜      Label the list, forcing it to take a concrete value.
    +     Take its sum.

I'm not entirely sure why ~× only produces lists with elements above 1, but it seems to do so, which works great in this challenge.

Answer 2

Gaia, 9 14 13 bytes

Bug fixed at the cost of 5 bytes thanks to Jonathan Allan, then 1 byte golfed.

ḍfḍ¦e¦Π¦¦Σ¦ul

Try it online! or try as a test suite

Explanation

ḍ              Prime factors
 f             Permutations
  ḍ¦           Get the partitions of each permutation
    e¦         Dump each list of partitions (1-level flatten the list)
      Π¦¦      Product of each partition
         Σ¦    Sum each group of products
           u   Deduplicate
            l  Length

Answer 3

Jelly,  11 15  14 bytes

+4 bytes fixing up a bug (maybe a better way?)
-1 byte by abusing symmetry

ÆfŒ!ŒṖ€ẎP€S€QL

A monadic link taking and returning positive integers

Try it online! or see a test suite

How?

Updating...

ÆfŒ!ŒṖ€ẎP€S€QL - Link: number, n      e.g. 30
Æf             - prime factors of n        [2,3,5]
  Œ!           - all permutations          [[2,3,5],[2,5,3],[3,2,5],[3,5,2],[5,2,3],[5,3,2]]
    ŒṖ€        - all partitions for €ach   [[[[2],[3],[5]],[[2],[3,5]],[[2,3],[5]],[[2,3,5]]],[[[2],[5],[3]],[[2],[5,3]],[[2,5],[3]],[[2,5,3]]],[[[3],[2],[5]],[[3],[2,5]],[[3,2],[5]],[[3,2,5]]],[[[3],[5],[2]],[[3],[5,2]],[[3,5],[2]],[[3,5,2]]],[[[5],[2],[3]],[[5],[2,3]],[[5,2],[3]],[[5,2,3]]],[[[5],[3],[2]],[[5],[3,2]],[[5,3],[2]],[[5,3,2]]]]
       Ẏ       - tighten                   [[[2],[3],[5]],[[2],[3,5]],[[2,3],[5]],[[2,3,5]],[[2],[5],[3]],[[2],[5,3]],[[2,5],[3]],[[2,5,3]],[[3],[2],[5]],[[3],[2,5]],[[3,2],[5]],[[3,2,5]],[[3],[5],[2]],[[3],[5,2]],[[3,5],[2]],[[3,5,2]],[[5],[2],[3]],[[5],[2,3]],[[5,2],[3]],[[5,2,3]],[[5],[3],[2]],[[5],[3,2]],[[5,3],[2]],[[5,3,2]]]
        P€     - product for €ach          [[30],[6,5],[10,3],[2,3,5],[30],[10,3],[6,5],[2,5,3],[30],[6,5],[15,2],[3,2,5],[30],[15,2],[6,5],[3,5,2],[30],[10,3],[15,2],[5,2,3],[30],[15,2],[10,3],[5,3,2]]
               -   ...this abuses the symmetry saving a byte over P€€
          S€   - sum €ach                  [30,11,13,10,30,13,11,10,30,11,17,10,30,17,11,10,30,13,17,10,30,17,13,10][10,17,11,30,10,17,13,30,10,13,11,30,10,13,17,30,10,11,13,30,10,11,17,30]
            Q  - de-duplicate              [30,11,13,10,17]
             L - length                    5

Answer 4

Python 2, 206 bytes

k=lambda n,i=2:n/i*[k]and[k(n,i+1),[i]+k(n/i)][n%i<1]
def l(t):
 r=[sum(t)]
 for i,a in enumerate(t):
    for j in range(i+1,len(t)):r+=l(t[:i]+[a*t[j]]+t[i+1:j]+t[j+1:])
 return r
u=lambda n:len(set(l(k(n))))

Try it online!

Explanation

    # Finds the prime factors
k=lambda n,i=2:n/i*[k]and[k(n,i+1),[i]+k(n/i)][n%i<1]
    # Function for finding all possible numbers with some repetition
def l(t):
    # Add the current sum
 r=[sum(t)]
    # For each number in the current factors
 for i,a in enumerate(t):
    # For all numbers further back in the current factors, find all possible numbers when we multiply together two of the factors
    for j in range(i+1,len(t)):r+=l(t[:i]+[a*t[j]]+t[i+1:j]+t[j+1:])
 return r
    # Length of set for distinct elements
u=lambda n:len(set(l(k(n))))

Answer 5

Mathematica, 110 bytes

If[#==1,1,Length@Union[Tr/@Select[Array[f~Tuples~{#}&,Length[f=Rest@Divisors[s=#]]]~Flatten~1,Times@@#==s&]]]&

Answer 6

JavaScript (ES6) 107 bytes

f=(n,o,s=0,i=2,q=n/i)=>(o||(o={},o[n]=t=1),i<n?(q>(q|0)|o[e=s+i+q]||(o[e]=t+=1),f(q,o,s+i),f(n,o,s,i+1)):t)

Ungolfed:

f=(n,                                 //input
   o,                                 //object to hold sums
   s=0,                               //sum accumulator
   i=2,                               //start with 2
   q=n/i                              //quotient
  )=>(
  o||(o={},o[n]=t=1),                 //if first call to function, initialize o[n]
                                      //t holds the number of unique sums
  i<n?(                               //we divide n by all numbers between 2 and n-1
    q>(q|0)|o[e=s+i+q]||(o[e]=t+=1),  //if q is integer and o[s+i+q] is uninitialized,
                                      //... make o[s+i+q] truthy and increment t
    f(q,o,s+i),                       //recurse using q and s+i
    f(n,o,s,i+1)                      //recurse using n with the next i
  ):t                                 //return t
)

Test cases:

f=(n,o,s=0,i=2,q=n/i)=>(o||(o={},o[n]=t=1),i<n?(q>(q|0)|o[e=s+i+q]||(o[e]=t+=1),f(q,o,s+i),f(n,o,s,i+1)):t)

s= [];

for(i = 1; i <= 103 ; i++) {
  s.push(f(i));
}

console.log(s.join`, `)

To verify that the function calculates the correct sums, we can output the keys of the object instead of t:

f=(n,o,s=0,i=2,q=n/i)=>(o||(o={},o[n]=t=1),i<n?(q>(q|0)|o[e=s+i+q]||(o[e]=t+=1),f(q,o,s+i),f(n,o,s,i+1)):Object.keys(o))

console.log(f(24));  //9, 10, 11, 14, 24

Answer 7

Python 3, 251 bytes

lambda n:1 if n==1else len(set(sum(z)for z in t(f(n))))
f=lambda n:[]if n==1else[[i]+f(n//i)for i in range(2,n+1)if n%i==0][0]
t=lambda l:[l] if len(l)==1else[[l[0]]+r for r in t(l[1:])]+[r[:i]+[l[0]*e]+r[i+1:]for r in t(l[1:])for i,e in enumerate(r)]

Try it online!

The design is basic:

1. factorize n into its prime factors (a prime factor may appear several times : 16 -> [2,2,2,2]). That's the function f.

2. compute the partitions of the list of prime factors, and multiply the factors in each partition. The partitions are found as in https://stackoverflow.com/a/30134039, and the products are computed on the fly. That's the function t.

3. The final function gets the products of each partition of n and sums them, the get the number of different values.

The result for 2310=2*3*5*7*11 is 49.

EDIT : Maybe needs fix, but I don't have time to look at it now (I'm in a hurry). Hint: is the result correct for 2310=2*3*5*7*11 ? I dont' think so.

EDIT2 : Huge fix. See above. Previous (buggy) version was: Try it online!

f computes the factors (, with a (0, n) instead of (1, n) as first element.

The lambda splits each factor in "sub-factors" and sums the those "sub-factors".