Sum of Consecutive Numbers

Any integer can be written as the sum of two or more positive consecutive integers. By definition, these are called polite numbers. The only impolite numbers, or numbers that cannot be written as the sum or two or more positive consecutive integers are powers of 2, and the only powers of two between 1000 and 3000 are 1024 and 2048, so the answer is 1024+2048, or 3072

Any ${2}^{a_n}$ with the exponent is non negative integer, can't be written as 2 or more consecutive numbers.

Proof: the integer $2$ can't be written as two consecutive numbers. So so does $4, 8, ...$ Look here: a. All odd numbers can be written as 2 or more consecutive numbers. Let me tak examples, for example $k$ and $k+1$ is the consecutive numbers. when we add them, we use our parity identity. If $k$ is even, $k+1$ is odd, vice versa. b. Most even numbers, but not all of them can be written as 3 or more consecutive numbers. We can't write them as 2 consecutive numbers since the parity identity said that the consecutive number addition outcomes the odd. That is why I say that to make even numbers, we have to write at least 3 or more consecutive numbers.

Why the power of 2 can't be expressed as two consecutive numbers?

Any number, except the power of two has at least one odd prime factor and 2. I'll take an example. if $k$ is even, when we factorize them, the prime factor must be 2 and odd prime number For the power of two, we can't since the prime factor of the power of 2 is only 2. Thus, we can't express as consecutives. QED

Assume B is the number that can be factored by the consecutive sequence: B=A+A+1+A+2+...+A+n-1=nA+(1+2+...+n-1) $nA+\frac{n(n-1)}{2}=B \Rightarrow A=\frac{B}{n} - \frac{n-1}{2}$ So: $A=2^{10}, 2^{11}$ are two numbers found. $Result=\boxed{3072}$