Question

The largest set of distinct odd numbers sum equal to n

Answer 1

As stated in another answer, the product can be made arbitrarily large. However, I will impose some rules where this would not be the case:

The number must be integers

The numbers must be positive

Now, I thought about which is better, less bigger numbers, like {51,49}, or a lot of smaller numbers. Then, I thought that if there would be no restriction that you could not use the same number twice, you could use just 50 2s, whose product would then be 250. So it seems like more smaller numbers would be better. Since the numbers cannot repeat, and seeing that 91=1+2+3+4+5+6+7+8+9+10+11+12+13, I played around with things and got {2,3,5,6,7,8,9,10,11,12,13,14}, whose sum is 100, and whose product is 21,794,572,800=14!/4. I am not positive that this is the right answer using my restrictions, so if anyone can get a bigger number, please let me know.