Set of distinct odd positive integers such that $\left(\frac{1}{O_{1}}+\frac{1}{O_{2}}+\frac{1}{O_{3}}+...+\frac{m+1}{O_{n}}\right)=1$

Question

I am looking for a set of distinct odd positive integers $3\leq{O_1}<O_2<O_3<...<O_n$ such that

$$\frac{1}{O_{1}}+\frac{1}{O_{2}}+\frac{1}{O_{3}}+...+\frac{m+1}{O_{n}}=1$$ such that $m\in\mathbb{N}$ and $1\leq{m}\leq5$, and other than the simple sets $S=\{3\}$, $m=2$ and $S=\{5\}$, $m=4$.

Is it possible such set of distinct odd positive integers? And could you give me an example?

Thanks in advance!

Answer 1

Comment:

You are write and it can be shown by argument. I checked this by following program in Python:

 for a in range (3, 200, 2):
 
   for b in range (3, 200, 2):
 
     for c in range (5, 200, 2):

       for d in range (9, 200, 2):
              
         for e in range (11, 200, 2):
          
            x=a*b*c*d+a*c*d*e+a*b*d*e+b*c*d*e+a*b*c*e

            y=a*b*c*d*e

            z=x/y

            t=int(z)

            if (t*Y)==x:

               print a, b, c, d, e

And only two sets of solutions was resulted"

(a, b, c, d, e)= (3, 3, 5, 9, 45), (3, 3, 5, 15, 15)

If you replace 3 by other odds like 7, 5, 13 . . . in one command started with 3 you will get no result.