A003569 - OEIS

A003569

For n>0, a(n) = least positive number m such that 4^m == +1 or -1 (mod 2n + 1), with a(0) = 0 by convention.

0, 1, 1, 3, 3, 5, 3, 2, 2, 9, 3, 11, 5, 9, 7, 5, 5, 6, 9, 6, 5, 7, 6, 23, 21, 4, 13, 10, 9, 29, 15, 3, 3, 33, 11, 35, 9, 10, 15, 39, 27, 41, 4, 14, 11, 6, 5, 18, 12, 15, 25, 51, 6, 53, 9, 18, 7, 22, 6, 12, 55, 10, 25, 7, 7, 65, 9, 18, 17, 69, 23, 30, 7, 21, 37, 15, 12, 10, 13, 26, 33, 81, 10

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OFFSET

0,4

COMMENTS

Multiplicative suborder of 4 (mod 2n+1) = sord(4, 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..82.

Eric Weisstein's World of Mathematics, Multiplicative Order

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

MATHEMATICA

lpn[n_]:=Module[{m=1, pm}, pm=PowerMod[4, m, 2n+1]; While[pm!=1 && pm != 2n, m++; pm=PowerMod[4, m, 2n+1]]; m]; Join[{0}, Array[lpn, 90]] (* Harvey P. Dale, May 22 2016 *)

Suborder[k_, n_] := If[n > 1 && GCD[k, n] == 1, Min[MultiplicativeOrder[k, n, {-1, 1}]], 0];

a[n_] := Suborder[4, 2 n + 1];

a /@ Range[0, 100] (* Jean-François Alcover, Mar 21 2020, after T. D. Noe in A003558 *)

CROSSREFS

Sequence in context: A302141 A387128 A077924 * A333596 A066670 A282270

Adjacent sequences: A003566 A003567 A003568 * A003570 A003571 A003572

KEYWORD

easy,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

STATUS

approved