Suborder Function
The multiplicative suborder of a number 
(mod 
) is the least exponent 
such that 
(mod 
), or zero if no such 
exists. An 
always exists if 
and 
.
This function is denoted 
and can be implemented in the Wolfram Language as:
Suborder[a_,n_] := If[n>1&& GCD[a,n] == 1,
Min[MultiplicativeOrder[a, n, {-1, 1}]],
0
]
The following table summarizes 
for small values of 
and 
.
 |
OEIS |
 for  , 1, ... |
| 2 |
|
0, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 5, 0, 6, 0, ... |
| 3 |
A103489 |
0, 0, 1, 0, 1, 2, 0, 3, 2, 0, 2, 5, 0, 3, 3, ... |
| 4 |
|
0, 0, 0, 1, 0, 1, 0, 3, 0, 3, 0, 5, 0, 3, 0, ... |
| 5 |
A103491 |
0, 0, 1, 1, 1, 0, 1, 3, 2, 3, 0, 5, 2, 2, 3, ... |
REFERENCES:
Sloane, N. J. A. Sequences A103489 and A103491 in "The On-Line Encyclopedia of Integer Sequences."
Wolfram, S.; Martin, O.; and Odlyzko, A. M. "Algebraic Properties of Cellular Automata." Comm. Math. Phys. 93, 219-258, 1984.
