This proves that the number of sums is at least the same at the number of divisors, but it doesn't prove the equivalence. I.e. could there be other sums...? I think you need to be more rigorous in demonstrating the reversibility of each step.
Greg -----Original Message----- From: Riley [mailto:[EMAIL PROTECTED] Sent: Wednesday, February 11, 2004 8:16 PM To: [EMAIL PROTECTED] Subject: Number of odd divisors Hello All! After reading the kwiki on this subject provided by ton at http://terje2.perlgolf.org/~golf-info/a1227-equivalence.html, I think there is a more intuitive way to see that the number of ways to write a number as the sum of sequences of consecutive positive integers the same as the number of odd divisors of a number. If a number, y, is divisible by x, z times, then y can be expressed as x + x + ... + x, z times. We can subtract the appropriate amount from the left values, and increment the respective right values accordingly to rewrite this as (x - c'1) + (x - c'2) + ... + x + ... + (x + c'2) + (x + c'1) An example demonstrates this more clearly: 35 is divisible by 7, 5 times, so we can write: 35 = 7 + 7 + 7 + 7 + 7 and rewrite this to: 35 = (7-2) + (7-1) + 7 + (7+1) + (7+2), which is the same as: 35 = 5 + 6 + 7 + 8 + 9 For every odd divisor, there will be a corresponding summation. The reason even divisors don't allow this should be obvious: we need an odd number of components in order to be able to add the same amounts on the right that we subtracted from the left... -Riley (o0lit3) P.S. - Note the following example as well (to avoid confusion with negative numbers) 27 is divisible by 3, 9 times, so we can write: 27 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 and rewrite this to: 27 = (3-4) + (3-3) + (3-2) + (3-1) + 3 + (3+1) + (3+2) + (3+3) + (3+4), which is the same as: 27 = -1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 Removing all non-positive integers is easy because 0 is trivial and numbers bellow 0 will cancel with their corresponding positive conterparts... 27 = 2 + 3 + 4 + 5 + 6 + 7 ------------------------------------------------------------------------------ This message is intended only for the personal and confidential use of the designated recipient(s) named above. If you are not the intended recipient of this message you are hereby notified that any review, dissemination, distribution or copying of this message is strictly prohibited. This communication is for information purposes only and should not be regarded as an offer to sell or as a solicitation of an offer to buy any financial product, an official confirmation of any transaction, or as an official statement of Lehman Brothers. Email transmission cannot be guaranteed to be secure or error-free. Therefore, we do not represent that this information is complete or accurate and it should not be relied upon as such. All information is subject to change without notice.
