# Number of sums of factors

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 question so answers will be scored in bytes, with fewer bytes being better.

## OEIS sequence

OEIS A069016

• Suggested test case 240 – Jonathan Allan Aug 6 '17 at 18:52
• Since 36 has caused a lot of debate, I suggest it as a test case. – user41805 Aug 6 '17 at 19:00
• @WheatWizard 12 * 3 – Business Cat Aug 6 '17 at 19:05
• I have 2,2,3,3 -> 10, 2,6,3 -> 11, 2,2,9 -> 13, 12,3 -> 15, 2,18 -> 20, 36 -> 36 – Business Cat Aug 6 '17 at 19:07
• 36 should be 7 because (2*3)+(2*3)=12 should be in the list too. – Jonathan Allan Aug 6 '17 at 19:15

# Brachylog, 8 bytes

{~×≜+}ᶜ¹


Try it online!

## 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.

• It only produces lists with elements above 1 because otherwise there are infinitely many lists, which is often bad in challenges like these. – Fatalize Mar 5 '18 at 7:20

# Gaia, 914 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

• Can you provide a TIO link containing the corresponding outputs the numbers 1 through 36 inclusive? – user41805 Aug 6 '17 at 18:51
• This is exactly like the Jelly answer... – Erik the Outgolfer Aug 6 '17 at 18:53
• The OP says the output for 36 should be 5, not 6 – user41805 Aug 6 '17 at 18:56
• According to OEIS, 36 gives 7 instead of 5, but yours gives 6 – user41805 Aug 6 '17 at 19:26
• Apparently Gaia leaves out [6 6] – user41805 Aug 6 '17 at 19:27

# 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   [[[,,],[,[3,5]],[[2,3],],[[2,3,5]]],[[,,],[,[5,3]],[[2,5],],[[2,5,3]]],[[,,],[,[2,5]],[[3,2],],[[3,2,5]]],[[,,],[,[5,2]],[[3,5],],[[3,5,2]]],[[,,],[,[2,3]],[[5,2],],[[5,2,3]]],[[,,],[,[3,2]],[[5,3],],[[5,3,2]]]]
Ẏ       - tighten                   [[,,],[,[3,5]],[[2,3],],[[2,3,5]],[,,],[,[5,3]],[[2,5],],[[2,5,3]],[,,],[,[2,5]],[[3,2],],[[3,2,5]],[,,],[,[5,2]],[[3,5],],[[3,5,2]],[,,],[,[2,3]],[[5,2],],[[5,2,3]],[,,],[,[3,2]],[[5,3],],[[5,3,2]]]
P€     - product for €ach          [,[6,5],[10,3],[2,3,5],,[10,3],[6,5],[2,5,3],,[6,5],[15,2],[3,2,5],,[15,2],[6,5],[3,5,2],,[10,3],[15,2],[5,2,3],,[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


# 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))))

• 194 bytes – ovs Aug 6 '17 at 21:19

# Mathematica, 110 bytes

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


# 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

# 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]
t=lambda l:[l] if len(l)==1else[[l]+r for r in t(l[1:])]+[r[:i]+[l*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".

• -19 bytes. – notjagan Aug 7 '17 at 1:05
• -1 byte from @notjagan improvement – Halvard Hummel Aug 7 '17 at 6:03
• Thanks to @notjagan, but the initial code was so wrong... – jferard Aug 9 '17 at 15:42
• Thanks to @HalvardHummel, but same remark as above. – jferard Aug 9 '17 at 15:42