I was inspired in the section **B48** of Guy's book ([1]) for ask myself a question that arises when one consider the sequence for integers $m\geq 1$

$$\mathcal{S}_m:=\prod_{n=1}^m\frac{s_n+1}{s_n-1}\tag{1}$$
where $s_n$ denotes the sequence of semiprimes, thus $s_1=4$, $s_2=6$, $s_3=9\ldots$, that is *A001358* in the OEIS (see the Wikipedia *Semiprime*).

Seems that it is obvious that exists an integer $m_0$ for which $\forall m\geq m_0$ the denominator of $S_m$ (written as irreducible fraction) is an even integer, while that its numerator is odd, thus seems that the following conjecture is "obvious".

**Conjecture.** There exists an integer $m_0$ such that $\forall m\geq m_0$
$$\prod_{n=1}^m\frac{s_n+1}{s_n-1}$$
is never an integer.

What I am saying is that semiprimes as $9,33$ or $129$ add in the denominator of the corresponding $S_m$ powers of two with a great exponent. But I don't know how to get a rigorous proof from this heuristic. I've tested the conjecture for the first few dozens of semiprimes.

Question.What is/Is there a proof for the existence of an integer $m_0$ such that $\forall m\geq m_0$ $$\prod_{n=1}^m\frac{s_n+1}{s_n-1}$$ is never an integer?Many thanks.

I hope that this question isn't obvious, and good for this MathOverflow. Thanks all.

## References:

[1] Richard K. Guy, *Unsolved Problems in Number Theory*, Unsolved problems in Intuitive Mathematics Volume I, Second Edition, Springer-Verlag (1994).

Figurate Number. Invoking a formula due to Sitaramachandrarao (see the WikipediaEuler's totient function), then combining with the $\endgroup$ – user142929 Aug 4 at 8:08