Find all ways to represent a positive integer as the sum of two or more consecutive positive integers. Enter a number to calculate its consecutive-integer representations and see the formula used for each result.
Enter a positive integer to find sequences of consecutive positive integers whose sum equals that number.
A sum of consecutive integers is an expression formed by adding integers that follow one another without gaps. For example, 3 + 4 + 5 + 6 + 7 = 25 .
The integers in the sequence increase by exactly 1 each time. The sequence can therefore be written as start, start + 1, start + 2, ...
This calculator searches for representations using two or more consecutive positive integers. A number itself is not counted as a consecutive sum of length one.
Example: 21 can be written as 10 + 11 and also as 6 + 7 + 8 .
Therefore 21 has more than one representation as a sum of consecutive positive integers.
Suppose a number is represented as a sequence of count consecutive integers beginning with start. The sequence is:
The sum of these consecutive integers is:
Solving this equation for the starting value gives:
The calculator checks possible sequence lengths and determines whether the resulting starting value is a positive integer. If it is, the corresponding sequence is a valid representation.
Consider the number 15. It can be represented in several ways using consecutive positive integers.
The calculator identifies both representations and displays the corresponding sequence length, starting value and calculation formula.
Representing integers as sums of consecutive numbers is closely connected with elementary number theory. The possible representations depend on the factors and divisibility properties of the number.
For example, a sequence of a particular length is only possible when the starting-value formula produces a positive integer. This makes divisibility an important part of the calculation.
Consecutive-sum problems can therefore be studied alongside other number-theory topics such as prime factorization , greatest common divisors and least common multiples .
Summe.org provides several specialized tools for working with sums, sequences and mathematical series.
If you are studying the arithmetic structure of integers, you may also find the following Summe.org calculators useful:
Consecutive integers are whole numbers that follow one another with a difference of exactly 1, such as 4, 5, 6, 7 and 8.
This calculator finds representations of a positive integer as the sum of two or more consecutive positive integers.
No. Some positive integers do not have a representation using two or more consecutive positive integers. The available representations depend on the arithmetic properties of the number.
Yes. Some integers have several different representations. For example, 15 can be written as 7 + 8 and also as 4 + 5 + 6.
No. This calculator specifically searches for sequences containing at least two consecutive positive integers.
For a sequence of length count, the calculator uses the formula start = [num − count × (count − 1) / 2] / count . A result is accepted when the starting value is a positive integer.
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