recontre (sub-factorial or derangement) numbers

[wouter meeussen]

A000166 they are. Round[n! / E ], or !n for short.
Counts permutations without fixed points.

Now look at the GCD of n! and !n.
{1,1,2,3,4,5,18,7,8,81,10,11,36,13,14,495,16,17,486,209,260,63, ..
it is different from (n-1) only
for n= {1,7,10,13,16,19,20,21,22,23,25,27,28,31,34..
why so?

Why is Mod[n! , !n] equal to n!-2*!n (that's A055596) for n>3 ?
Why is GCD[n! , !(n+1)/n ] equal to
{1,1,3,1,1,3,1,1,9,1,11,3,1,1,33,1,1,27,11,13,3,11,1,3,1,11,9,1,1,363,1,1,429,1,.. with so many ones
and multiples of three? Where does the 'eleven-ness' come from?

One fool surely can ask more questions than a thousand gods can answer.

!W.