Consecutive Fun

First, lets say that when a number is expressed as a sum of consecutive integers, the number of integers that make it up is $k$ and the base number is $n$ .

For example, if $39=12+13+14$ , $12=n$ and $3=k$ .

So, for a integer $p$ , it can be expressed as $p=kn+\frac { k(k-1) }{ 2 }$ For example, substituting $12=n$ and $3=k$ gives $p=39$ .

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Proof that all odds can be expressed as a sum of 2 or more consecutive integers:

Any odd integer can be expressed as $2n+1=(n)+(n+1)$ , where $n$ is an integer.

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Let $p_{a}$ be the any integer not expressible as sums of 2 or more consecutive integers.

So now we have to revisit the formula: $p=kn+\frac { k(k-1) }{ 2 }$

Solving for $n$ gives: $n=\frac { p }{ k } -\frac { k-1 }{ 2 }$

Remember that $p_{a}$ has to be even?

$k$ has to be odd as if $k$ is even, $n$ would not be an integer due to the fact that $\frac { p }{ k } -\frac { k-1 }{ 2 }$ would not be an integer.

So now, if $k$ is odd, and that $k=2a+1$ the formula for $n$ becomes:

$n=\frac { p_{a} }{ 2a+1 } -\frac { 2a }{ 2 }$

For $p_{a}$ , $k$ has to be $1$ . If $p$ is divisible by $k$ where $k>1$ , $p$ is not $p_{a}$

So when $a>0$ , $p_{a}$ cannot be divisible by any odd number. Since any number is made up by product of primes which are odd, there is no such $p_{a}$ ...

Wait a minute, $2$ is even and IS a Prime! So, $p_{a}$ has to be made up of products of $2$

So $p_{a}$ is in the form of $2^{x}$ where $x$ is an integer!

The answer is thus $2^{6}=\boxed{64}$ !

The problem does not say anywhere that the integers must be positive integers, only that there must be two or more consecutive integers. Therefore, ANY whole number (even starting at 0) may be written as the sum of consecutive integers.

0 = -1+0+1

1= 0+1

2 = -1+0+1 + 2

3= -2+ -1+0+1 + 2 + 3

4 = -3 + -2 + -1+0+1 + 2 + 3 + 4

5 = -4 + -3 + -2 + -1+0+1 + 2 + 3 + 4 + 5

6= -5 + -4 + -3 + -2 + -1+0+1 + 2 + 3 + 4 + 5 + 6

7 = -6 + -5 + -4 + -3 + -2 + -1+0+1 + 2 + 3 + 4 + 5 + 6 + 7

8 = -7 + -6 + -5 + -4 + -3 + -2 + -1+0+1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

...

16 = -15 + -14 + -13 + ... + -1 + 0 + 1 + ... + 13+14+15+16

32 = -31 + -30 + ... + 30 + 31+ 32

...

and 64 = -63 + ... + 63 +64

128 = -127 + ... + 127 + 128