Suppose we have a
<a href="http://en.wikipedia.org/wiki/Bracelet_(combinatorics)">combinatorial bracelet</a> composed of natural numbers.

What is the number of different bracelets whose elements sum up to a previously fixed natural number N?

Also, are there any results if we add a constraint that the number of beads on the bracelet is always odd?

P.S. Any good upper bounds are also helpful.

(**EDITED** in the light of the comments below)