The integer 30 can be written as a sum of consecutive positive integers in three ways:
30 = 9+10+11 = 6+7+8+9 = 4+5+6+7+8.

Find the smallest positive integer which can be written as a sum of consecutive positive integers in 12 ways.

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The solution is the smallest number that has at least 13 odd divisors. Charlie's program shows that this number is 945=3x3x3x5x7 which has 16 odd divisors (including itself and 1), so if the trivial representation 945=945 is included, odd divisors and representations as sums of consecutive positive numbers are in one-to-one correspondence.

Posted by Richard on 2004-06-28 22:27:28