# Products taken over semiprimes

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).

• It is not clear to me why the sequence $\{s_n\}$ is particularly interesting in formulating a conjecture like this, rather than some other infinite sequence of positive integers. Aug 3 '19 at 13:10
• (1/2) Yes, @StanleyYaoXiao I am asking just about a sequence, because if with few computational evidence I ask about more sequences then the post doesn't seem serious. I thought, but I did not do the calculations for the sequence of square-free integers (the same product, specialization for squarefree integers, taken for $n\geq 2$), and I tried with variants of some of the best known sequences of figurate numbers, see the table in the encyclopedia MathWorld Figurate Number. Invoking a formula due to Sitaramachandrarao (see the Wikipedia Euler's totient function), then combining with the Aug 4 '19 at 8:08
• (2/2) convergence criteria for infinite products I know that $\prod_{n=2}^\infty\frac{n\varphi(n)+1}{n\varphi(n)-1}$ will be convergent, but I think that has no a nice closed-form. Thus this is an invitation if you, or some of your colleagues in MO want to study more variants (the problems that have a good mathematical content are those showed in Guy's book). With my computational evidence and mathematical experience, I could not propose a more interesting product. Many thanks again, as was said this is an invitation if some user want try to explore these products or variants of problems Aug 4 '19 at 8:08