# Integer-as-a-Sum Rewriter

## World's Simplest Integer Tool

This browser-based utility rewrites an integer as a sum of several smaller integers and prints the additive decomposition in the output. You can specify the number of addends in the sum and the number of sums in the options. You can also change the order of the terms, customize the plus sign symbol, and the sum separator and surrounding characters. Created by math nerds from team Browserling.

Tool Options

Rewrite an integer as a sum of
this many smaller integers.
Output this many different sums.
in integer sums.

#### Sum Symbols

Enter a custom plus symbol here.
Separate sums by this char.
Sum open symbol.
Sum close symbol.

small to large.
large to small.
random order.

## What Is an Integer-as-a-Sum Rewriter?

With this online application, you can express an integer as a sum of two or more other positive values. For example, the integer 10 can be written using two terms as 4 + 6, using three terms as 5 + 3 + 2, using four terms as 1 + 4 + 2 + 3, or more terms. The decomposition 4 + 6 is considered to be the same as 6 + 4 because addition is commutative. You can specify the number of addends that need to go in the sum, as well as choose if the integer 0 should be used as an addend. If the zero addend is disabled, then the integer n can be decomposed into a sum that has strictly from 1 to n addends (in case of n addends, they are all 1's). If you allow the use of the zero, then it can consist of any number of addends as you can add as many zeros to an integer as you want and it won't change its value. The larger the input integer, the more integer partitions there are. For example, the integer 12 can be rewritten as a sum of 3 summands in 19 different ways and the integer 120 can be represented as 1238 unique sums. To make sure you get quick results and the output isn't too long, we have limited the output to 100 sums. You can increase or decrease this count in the options. By default, the summands are displayed from the smallest to the largest (for example, 1 + 1 + 2 + 8) but you can change their order in the options and print them from the largest to the smallest (for example, 8 + 2 + 1 + 1) or even in random order (for example, 1 + 8 + 2 + 1). You can also change the "+" sign to any other symbol, print the sums one after another (and not in a column), separate them with custom symbols, and also surround the sums with custom open and close symbols so that the output could immediately be used in Python or JavaScript programming languages. For example, if the open and close symbols are "(" and ")", and the summands are separated by commas, you will get tuples such as "(1, 1, 2, 8)". Integerabulous!

## Integer-as-a-Sum Rewriter Examples

Click to try!
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In this example, we expand the integer 9 into a sum of three other integers, including the zero integer. It turns out that for this input integer there are twelve unique three-term sums. We print them one per line and display the summands in increasing order and place the regular plus symbol "+" between them.

9
0 + 0 + 9 0 + 1 + 8 0 + 2 + 7 0 + 3 + 6 0 + 4 + 5 1 + 1 + 7 1 + 2 + 6 1 + 3 + 5 1 + 4 + 4 2 + 2 + 5 2 + 3 + 4 3 + 3 + 3
Required options
These options will be used automatically if you select this example.
Rewrite an integer as a sum of
this many smaller integers.
Output this many different sums.
in integer sums.
Enter a custom plus symbol here.
Separate sums by this char.
Sum open symbol.
Sum close symbol.
small to large.
click me

### Sums without Zero

In this example, we disallow the use of zero-summand and generate all possible commutatively-unique sum combinations of the integer 12. Without using a zero, the number of partitions is 11, while using a zero, it would have reached 58. We replace the plus sign with the colon character and output all terms in decreasing order. We make the output comma-separated that prints all sums on the same line and enclose each sum in a pair of curly brackets.

12
{7:1:1:1:1:1}, {6:2:1:1:1:1}, {5:3:1:1:1:1}, {4:4:1:1:1:1}, {5:2:2:1:1:1}, {4:3:2:1:1:1}, {3:3:3:1:1:1}, {4:2:2:2:1:1}, {3:3:2:2:1:1}, {3:2:2:2:2:1}, {2:2:2:2:2:2}
Required options
These options will be used automatically if you select this example.
Rewrite an integer as a sum of
this many smaller integers.
Output this many different sums.
in integer sums.
Enter a custom plus symbol here.
Separate sums by this char.
Sum open symbol.
Sum close symbol.
large to small.
click me

### Random Integer Sums

In this example, we split the integer 100 into random sums with eight addends. There are more than one million partitions for this input specification and it's too large to compute. To quickly get usable results, we have limited the output to just 15 values. They are printed in a vertical column as the sum separator symbol is set to "\n".

100
1+1+92+1+1+1+2+1 1+1+4+65+1+1+26+1 74+1+1+18+1+3+1+1 60+13+1+4+1+1+1+19 1+58+2+35+1+1+1+1 25+1+3+6+29+8+9+19 1+18+30+34+4+2+1+10 10+4+1+1+56+20+1+7 3+1+85+1+1+1+7+1 5+1+1+88+1+1+1+2 1+41+8+1+1+1+4+43 1+61+1+10+3+16+1+7 40+1+1+1+4+1+9+43 3+1+1+1+91+1+1+1 1+10+51+2+2+6+5+23
Required options
These options will be used automatically if you select this example.
Rewrite an integer as a sum of
this many smaller integers.
Output this many different sums.
in integer sums.
Enter a custom plus symbol here.
Separate sums by this char.
Sum open symbol.
Sum close symbol.
random order.
click me

### Sums for a Large Integer

In this example, we enter a huge integer equal to nine billion and decompose it into sums of four terms. We limit the number of sums in the output to six, as there are nearly infinitely many of them and it would take a very long time to calculate them all. The sums are randomly generated each time when you click on the example and in each case, adding all the terms together produces 9,000,000,000.

9000000000
1711990448 + 1620447777 + 4513236069 + 1154325706 65092647 + 7195252690 + 1087127447 + 652527216 75025523 + 445960474 + 7275699612 + 1203314391 8859019917 + 27328377 + 36962265 + 76689441 3859074492 + 3054860384 + 1950209345 + 135855779 6999192102 + 502145916 + 975841729 + 522820253
Required options
These options will be used automatically if you select this example.
Rewrite an integer as a sum of
this many smaller integers.
Output this many different sums.
in integer sums.
Enter a custom plus symbol here.
Separate sums by this char.
Sum open symbol.
Sum close symbol.
random order.
click me

In this example, the number of addends is greater than the input integer. In order to produce 8 terms that add up to the integer 6, some of the addends must be equal to 0. To allow this condition, we have enabled the "Include Zero" option, which creates two or more redundant zero terms in the sums. We print the sums in parentheses, separate terms by the comma character, and place zeros at the beginning of sums.

6
(0, 0, 0, 0, 0, 0, 0, 6) (0, 0, 0, 0, 0, 0, 1, 5) (0, 0, 0, 0, 0, 0, 2, 4) (0, 0, 0, 0, 0, 0, 3, 3) (0, 0, 0, 0, 0, 1, 1, 4) (0, 0, 0, 0, 0, 1, 2, 3) (0, 0, 0, 0, 0, 2, 2, 2) (0, 0, 0, 0, 1, 1, 1, 3) (0, 0, 0, 0, 1, 1, 2, 2) (0, 0, 0, 1, 1, 1, 1, 2) (0, 0, 1, 1, 1, 1, 1, 1)
Required options
These options will be used automatically if you select this example.
Rewrite an integer as a sum of
this many smaller integers.
Output this many different sums.
in integer sums.
Enter a custom plus symbol here.
Separate sums by this char.
Sum open symbol.
Sum close symbol.
small to large.

### Pro tipsMaster online integer tools

You can pass input to this tool via ?input query argument and it will automatically compute output. Here's how to type it in your browser's address bar. Click to try!

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