I searched for prime numbers of the form $$(\prod_{j=1}^n \phi(j))+1$$ where $\phi(n)$ is the totient function. I wonder whether the function $$f(n):=\prod_{j=1}^n \phi(j)$$ has a nice closed form using factorials, lcm's or whatsoever.
For those who are interested : $f(n)+1$ is prime for the following positive integers $n\le 1000$ : $$[1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 34, 43, 85, 162, 173, 538, 552]$$ Curious side-note : $f(43)+1$ is a $43$ digit-prime and $43$ is prime as well.
Does a nice formula for $f(n)$ exist ?
For an $n\times n$ matrix with $(i,j)$th entry as as $\gcd(i,j)$ the determinant is the product you mentioned. This is found in the book by Niven and Zuckerman "Number Theory"