For a positive integer $\ m\ $ define $\ k_m\ $ to be the smallest positive integer such that $$k_m\cdot m!\pm 1$$ form a twin-prime pair.
Has the sequence $(k_m)$ already been verified ?
The first $\ 30\ $ entries of the sequence are :
[4, 2, 1, 3, 2, 17, 7, 6, 3, 14, 29, 30, 48, 27, 9, 24, 12, 97, 78, 47, 71, 80, 55, 13, 57, 20, 81, 259, 108, 163]
I am particular interested in $\ k_{2022}\ $
UPDATE : hardmath found the twin prime $$1892020\cdot 2022!\pm 1$$ so we have $$k_{2022}\le 1\ 892\ 020$$