Let $y$ be the vector
[3145940416080, 8695630985278406400, 9452665808524931731200, 6185588188453502201
60400, 899721918320509411142400, 989694110152560352256640, 109966012239173372472
9600, 1237117637690700440320800, 1413848728789371931795200, 16494901835876005870
94400, 1979388220305120704513280]
It is derived from the vector $x$
[5, 6, 7, 8, 9, 10, 11, 16, 1047, 1138151, 3145940416080]
with the property $\sum_{j=1}^{11} \frac{1}{x_j}=1$
Because of $y(s)=\frac{lcm(x)^2}{x(s)}$ (the vector $y$ was sorted afterwards) , we have the following property :
If $k$ is a natural number, such that $k\times y[j]+1$ is prime for all $j=1,...,11$, then $\prod_{j=1}^{11} k\times y[j]+1$ is a Carmichael-number.
Which is the smallest $k$, such that $k\times y[j]+1$ is prime for $j=1,...,11$ ?