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.