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Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers

Some natural numbers can be expressed as a sum of consecutive natural numbers in more than one way. For example, $7$ can get expressed both as $7$, and $(3+4).$ In terms of a sum of consecutive numbers, $4$ and $8$ can only get expressed as $4$, and $8$ respectively. Call such numbers consecutive-primes. How many consecutive-primes exist? Given all previous consecutive-primes, is there a way to compute the next consecutive-prime?

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Doug Spoonwood
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Only powers of 2 have exactly 1 odd divisor.

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Dan Brumleve
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  • Huh? What do you mean by "odd divisor"? I mean, 3 only has one divisor (discounting 1), which is odd, if by "odd divisor" you mean a divisor which is odd. Also, how does the notion of "divisor" end up relating here? – Doug Spoonwood May 02 '12 at 12:38
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    @DougSpoonwood Can you see how to use an odd divisor to create a sum of consecutive numbers? Hint: 3=0+1+2 – Mark Bennet May 02 '12 at 12:49