Which numbers cannot be a sum of consecutive numbers?

Question

Nine is 4 + 5 and also 2 + 3 + 4. Fifteen is 7 + 8. Ten is 1 + 2 + 3 + 4. But 8 cannot be written as a sum of two or more consecutive positive whole numbers, and neither can 16.

Which positive whole numbers cannot be written as a sum of two or more consecutive positive whole numbers? Prove your characterisation in both directions.

Accepted Answer

Exactly the powers of two: 1, 2, 4, 8, 16, ... A consecutive sum always has an odd factor bigger than 1. Take k consecutive numbers starting at a, with k at least 2. Their sum is k × (first + last)/2 = k × (2a + k minus 1)/2. - If k is odd, then k itself is an odd factor of the sum, bigger than 1. - If k is even, then 2a + k minus 1 is odd, and it is at least 2 × 1 + 2 minus 1 = 3. The sum equals (k/2) × (2a + k minus 1), so this odd number bigger than 1 divides the sum. A power of two has no odd factor bigger than 1, so no power of two is a consecutive sum. Every other number is a consecutiv…

Which numbers cannot be a sum of consecutive numbers?

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