[seqfan] Catastrophe theory in number theory

[Artur]

Dear Seqfans,

I was find some interesting relations during examination sequences of 
the kind

Smallest k such that m^k mod k = n

see catalogue of this sequences in comment to A178194
http://www.research.att.com/~njas/sequences/A178194

some rules which I was observed are:
1) existed sequences of succesive numbers m when also n are succesive numbers when k is exploding big number in comparission with previous numbers (I'm naming this catastrophe)
eg. set of m n where catastrophe occured 
m n
36 19
37 20
38 21
39 22
40 23
is this coincidence or rule ?
2) existed some numbers where first weak catastrophe and first strong catastrophe occured for these same values e.g.
m n_weak n_strong
21 16 42
41 16 42

I will be greatfull for any ideas, comments or proofs.

Best wishes
Artur Jasinski