Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers How do I prove that any even integer $n \neq 2^k$ is expressible as a sum of positive consecutive integers (more than 2 positive consecutive integer)?
For example: 
14 = 2 + 3 + 4 + 5
84 = 9 + 10 + ... + 15

n = sum (k + k+1 + k+2 + ...) 
n ≠ 2^k


The argument of @Arturo Magidin that powers of $2$ are not representable can be extended to a "formula" for the number of representations of any positive integer.  For no good reason, we use slightly different notation.
Let $w$ be a positive integer.  We look for integers $m\ge 0$, $n>m$ such that
$$w=(1+2+\cdots +n)-(1+2+\cdots +m).$$
With a little manipulation, we can rewrite this as
$$2w=(n-m)(n+m+1).$$
Temporarily, we allow $n-m=1$, even though this corresponds to expressing $w$ as the "sum" of one integer.
The numbers $n-m$ and $n+m+1$ are of opposite parity and have product $2w$. 
Note also that $n-m<n+m+1$.
Now take any two numbers $x$, $y$ of opposite parity whose product is $2w$. 
Set $n-m=x$ and $n+m+1=y$. We can solve for $n$ and $m$, and obtain a representation of $w$ as a sum of consecutive integers (again, possibly only one such integer.)
We obtain such $x$, $y$ of opposite parity as follows.  Let $2w=2^k q$ where $q$ is odd.  Express $q$ as a product of two positive integers $u$ and $v$, not necessarily distinct. Multiply one of $u$ or $v$ (say $v$) by $2^k$, and let $x$ be the smaller of $u$ and $2^k v$, and  $y$ the larger.
So the number of representations of $w$ as a sum of consecutive positive integers is $d(q)$, the number of divisors of $q$.  If we do not want to allow the trivial representation of $w$ as the short sum $w$, the number of representations is $d(q)-1$.  
If $w$ is a power of $2$, then $q=1$, and therefore there are no representations.  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 is 
$$(a_1+1)(a_2+1)\cdots(a_k+1)-1.$$
In particular, if $w$ is not a power of $2$, there is at least one non-trivial representation.  
Added: The OP has now excluded the possibility of $2$ consecutive integers.  Since the original question is about even $w$, that makes no difference. For odd $w$, we just remove  $1$ of the representations counted above. 
