A279185 Triangle read by rows: T(n,k) (n>=1, 0 <=k<=n-1) is the length of the period of the sequence obtained by starting with k and repeatedly squaring mod n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1

Offset

1,24

Comments

Fix n. Start with k (0 <= k <= n-1) and repeatedly square and reduce mod n until this repeats; T(n,k) is the length of the cycle that is reached.

A279186 gives maximal entry in each row.

A037178 gives maximal entry in row p, p = n-th prime.

A279187 gives maximal entry in row c, c = n-th composite number.

A279188 gives maximal entry in row c, c = prime(n)^2.

A256608 gives LCM of entries in row n.

A256607 gives T(2,n).

Links

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Example

The triangle begins:
1,
1,1,
1,1,1,
1,1,1,1,
1,1,1,1,1,
1,1,1,1,1,1,
1,1,2,2,2,2,1,
1,1,1,1,1,1,1,1,
1,1,2,1,2,2,1,2,1,
1,1,1,1,1,1,1,1,1,1,
1,1,4,4,4,4,4,4,4,4,1,
...
For example, if n=11 and k=2, repeatedly squaring mod 11 gives the sequence 2, 4, 5, 3, 9, 4, 5, 3, 9, 4, 5, 3, ..., which has period T(11,2) = 4.

Maple

A279185 := proc(k, n) local g, y, r;
  if k = 0 then return 1 fi;
  y:= n;
  g:= igcd(k, y);
  while g > 1 do
     y:= y/g;
     g:= igcd(k, y);
  od;
  if y = 1 then return 1 fi;
  r:= numtheory:-order(k, y);
  r:= r/2^padic:-ordp(r, 2);
  if r = 1 then return 1 fi;
  numtheory:-order(2, r)
end proc:
seq(seq(A279185(k, n), k=0..n-1), n=1..20); # Robert Israel, Dec 14 2016

Mathematica

T[n_, k_] := Module[{g, y, r}, If[k == 0, Return[1]]; y = n; g = GCD[k, y]; While[g > 1, y = y/g; g = GCD[k, y]]; If[y == 1, Return[1]]; r = MultiplicativeOrder[k, y]; r = r/2^IntegerExponent[r, 2]; If[r == 1,  Return[1]]; MultiplicativeOrder[2, r]];
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

Crossrefs

Cf. A037178, A256607, A256608, A279186, A279187, A279188.

See also A141305, A279189, A279190, A279191, A279192.

Sequence in context: A230415 A101080 A130836 * A161385 A152907 A078786

Adjacent sequences: A279182 A279183 A279184 * A279186 A279187 A279188

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 14 2016

Status

approved