# In how many ways can a number be expressed as a sum of consecutive numbers?

All the positive numbers can be expressed as a sum of one, two or more consecutive positive integers. For example 9 can be expressed in three such ways, 2+3+4, 4+5 or 9. In how many ways can a number be expressed as a sum of consecutive numbers?

Here for 9 answer is 3, for 10 answer is 3, for 11 answer is 2.

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@BhavikAmbani, this question is like asking if n is a fibonacci number. The solution can be calculated, but it can't be represented by a simple formula, unless you're saying something trivial like $f(n) \Leftrightarrow f(n - 1) + f(n - 2)$ –  Neil May 2 '12 at 10:37
@BhavikAmbani: Many things are counted without explicit formulas, like the prime counting function. However, this question is equivalent to the Diophantine(ish?) problem of counting the integer solutions $(a,b)$ with $a<b$ to $$(b+a)(b-a+1)=2n$$ for $n$ given but unknown. –  anon May 2 '12 at 10:49
@anon Very true, at least provide the basic logic for that –  Bhavik Ambani May 2 '12 at 10:58
Please do not engage in excessive discussions in the comments. If you wish to discuss, please use our chatroom instead. I'll be cleaning up the comments here which are wandering a bit off-topic in a minute or so. –  Willie Wong May 2 '12 at 10:59
@BhavikAmbani: (fixed bad typo, earlier now deleted comment) The following is a formula quoted from the post I referred to. For the proof please see the post. If $$w=2^{a_0}p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k},$$ where the $p_1,p_2,\dots,p_k$ are distinct odd primes, then the number of non-trivial representations of $w$ is $$(a_1+1)(a_2+1)\cdots(a_k+1).$$ –  André Nicolas May 2 '12 at 17:31

Here's one more way to calculate this, from my answer to this question on another site:

An integer $n$ is expressible as the sum of $m$ consecutive positive integers if and only if either:

• $m$ is odd and $\frac nm$ is an integer, or
• $m$ is even and $\frac nm + \frac12$ is an integer,

and $\frac nm \ge \frac m2$ (or else some of the integers in the sum would be zero or negative).

These conditions follow from the fact that the sum of an arithmetically increasing sequence of $m$ numbers equals $m$ times the mean of the numbers.

The last condition can be rewritten as $m \le \sqrt{2n}$. Thus, it's sufficient to iterate over all integers $m$ from $1$ to $\lfloor \sqrt{2n} \rfloor$ and check whether $\frac nm + \frac m2 + \frac12$ is an integer.

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Very Good Answer.. +1 for this –  Bhavik Ambani May 3 '12 at 5:06
what if instead of n , n! is given ? answer will be n-2 ? –  pola sai ram May 17 at 5:31

A sum of consecutive numbers is a difference of triangular numbers. The paper below gives a solution for the case of nonconsecutive triangular numbers.

Nyblom, M. A. On the representation of the integers as a difference of nonconsecutive triangular numbers. Fibonacci Quart. 39 (2001), no. 3, 256–263.

The main result is that the number of distinct representations of a nonzero integer $m$ as a difference of nonconsecutive triangular numbers is given by $d−1$, where $d$ is the number of odd divisors of $m$.

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An equivalent statement to that main result is given in the comments on A001227, although without reference. –  Peter Taylor May 2 '12 at 13:59
Also explained in an answer by André Nicolas. –  lhf May 2 '12 at 14:14

Fix $k$. Is there a way that a number $N$ can be written in more than one way as a sum of $k$ consecutive number? Certainly not because $$a+(a+1)+\cdots+(a+k-1)\neq b+(b+1)+\cdots+(b+k-1)$$ if $a\neq b$. On the other hand $N$ is the sum of $k$ consecutive number if and only if $N$ is the form $$N=\frac12\left[(n+k)(n+k+1)-n(n+1)\right]$$ for some $n$. Does that help?

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Thanks a lot for your efforts, but this does not make any solution for me –  Bhavik Ambani May 2 '12 at 11:05
It was not intended as a solution, but as a hint/help. –  Andrea Mori May 2 '12 at 12:30
But sorry I couldn't get your hint till now. –  Bhavik Ambani May 2 '12 at 12:31

Factorise the number and find the number of odd factors . Total number of odd factors (except 1) is the answer.

Express N in terms of prime factors

$N = a^p . b^q . c^r$

If a = 2 . Number of odd factors = (q+1)(r+1) - 1 . Note : 1 is subtracted because 1 cannot be answer as consecutive terms means greater than 1 term.

For eg.

$100 = 2^2 . 5^2$

So Number of odd factors = (2+1) - 1 = 2 = Number of ways of writing 100 as sum of 2 or more consecutive integers . They are
18, 19, 20, 21, 22

9,10,11,12,13,14,15,16

 Number of ways of writing N as sum of consecutive positive integers is Number of odd factors in that number (except 1). 

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I thought of this same question several months ago during one of my classes and I worked out the solution during my lunch break, the same sort of argument could be used to find the number of representations of $n$ in any arithmetic sequence modulo a positive integer. I got that if $S(n)$ denotes the number of representations of $n$ as a sum of successive natural numbers with $n\ge 1$ then that:

$$S(n)=d(\frac{n}{2^{v_2(n)}})$$

Where $v_2(n)$ is the $2$-adic order of $n$, what I did was used the fact that:

$$\sum_{\substack{a^2+ab=n\\(a,b)\in \mathbb{N^2}}}f(a,b)=\sum_{\substack{b=\frac{n}{a}-a\\(a,b)\in \mathbb{N^2}}}f(a,b)=\sum_{\substack{d\mid n\\d<\sqrt{n}}}f(d,\frac{n}{d}-d)$$

To rewrite: $$S(n)=\sum_{\substack{a+(a+1)+(a+2)+\dots +(b-1)+b=n\\b\ge a\\(a,b)\in \mathbb{N^2}}}1=\sum_{\substack{(a+b)(a-b+1)=2n\\b\ge a\\(a,b)\in \mathbb{N^2}}}1=\sum_{\substack{a^2-b^2+a+b=2n\\b\ge a\\(a,b)\in \mathbb{N^2}}}1$$

And then simplified the resulting sum by swapping the summation indices several times and by setting $b=a-1+k$ with $k\in \mathbb{N}$ since we have that $b\ge a$.

This was the proof I scribbled down, where I used $\chi_2$ to denote the Dirichlet character modulo $2$.

Sorry if it's kind of messy:

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Good one, +1 for the nice explaination :) –  Bhavik Ambani Mar 20 '14 at 9:06