A003570 For n>0, a(n) = least positive number m such that 8^m == +1 or -1 (mod 2n + 1), with a(0) = 0 by convention.

0, 1, 2, 1, 1, 5, 2, 4, 4, 3, 2, 11, 10, 3, 14, 5, 5, 4, 6, 4, 10, 7, 4, 23, 7, 8, 26, 20, 3, 29, 10, 2, 2, 11, 22, 35, 3, 20, 10, 13, 9, 41, 8, 28, 11, 4, 10, 12, 8, 5, 50, 17, 4, 53, 6, 12, 14, 44, 4, 8, 55, 20, 50, 7, 7, 65, 6, 12, 34, 23, 46, 20, 14, 14, 74, 5, 8, 20, 26, 52, 11, 27, 20, 83

Offset

0,3

Comments

Multiplicative suborder of 8 (mod 2n+1) = sord(8, 2n+1). - Harry J. Smith, Feb 11 2005

References

H. Cohen, Course in Computational Algebraic Number Theory, Springer, 1993, p. 25, Algorithm 1.4.3

Links

Table of n, a(n) for n=0..83.

Eric Weisstein's World of Mathematics, Multiplicative Order.

S. Wolfram, Algebraic Properties of Cellular Automata (1984), Appendix B.

Example

a(1) = 1 since 8^1 = 8 == -1 (mod 3).
a(2) = 2 since 8^2 = 64 == -1 (mod 5).

Mathematica

Suborder[k_, n_] := If[n > 1 && GCD[k, n] == 1, Min[MultiplicativeOrder[k, n, {-1, 1}]], 0];
a[n_] := Suborder[8, 2 n + 1];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 21 2020, after T. D. Noe in A003558 *)

Crossrefs

Sequence in context: A259703 A316996 A169589 * A011281 A300731 A100398

Adjacent sequences: A003567 A003568 A003569 * A003571 A003572 A003573

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Harry J. Smith, Feb 11 2005

Edited by N. J. A. Sloane, May 22 2008 at the suggestion of Jeremy Gardiner

Status

approved