Consider n=45; then

$$1+2+3+4+5+6+7+8+9=45$$ $$5+6+7+8+9+10=45$$ $$7+8+9+10+11=45$$ $$14+15+16=45$$ $$22+23=45$$

Question: how to find all representations of a given positive integer n as a sum of consecutive positive integers smaller than n?

It is sufficient (and concise) to provide only the start and end values of each sequence, i.e. {1, 9}, {5, 10} etc.

Consider n=45; then

$$1+2+3+4+5+6+7+8+9=45$$ $$5+6+7+8+9+10=45$$ $$7+8+9+10+11=45$$ $$14+15+16=45$$ $$22+23=45$$

Question: how to find all representations of a given positive integer n as a sum of consecutive positive integers smaller than n with an efficient Mathematica implementation? A brute-force method is straightforward, but it's likely to fail for big n, so a more clever idea is required.

It is sufficient (and concise) to provide only the start and end values of each sequence, i.e. {1, 9}, {5, 10} etc.