It is well known that $$(6n+1)(12n+1)(18n+1)$$ is a Carmichael number, if all factors are prime. I found the polynomials $$(6n+1)(12n+1)(18n+1)(36n+1)$$ $$(18n+1)(36n+1)(108n+1)(162n+1)$$ $$(20n+1)(80n+1)(100n+1)(200n+1)$$ $$(12n+1)(24n+1)(36n+1)(72n+1)(144n+1)$$
with the same property.
Questions : Is there an efficient way to construct more such polynomials (without brute force) ?
Does such a polynomial with degree $d$ exist for all $d\ge 3$ ?
Besides the multiples of the vector $[6,12,18]$, does a vector $[a,b,c]$ exists, such that $(an+1)(bn+1)(cn+1)$ is such a polynomial ?