A343721 a(n) is the number of starting residues r modulo n from which repeated iterations of the mapping r -> r^2 mod n reach a fixed point.

1, 2, 3, 4, 5, 6, 3, 8, 5, 10, 3, 12, 5, 6, 15, 16, 17, 10, 3, 20, 9, 6, 3, 24, 9, 10, 11, 12, 5, 30, 3, 32, 9, 34, 15, 20, 5, 6, 15, 40, 9, 18, 3, 12, 25, 6, 3, 48, 9, 18, 51, 20, 5, 22, 15, 24, 9, 10, 3, 60, 5, 6, 15, 64, 25, 18, 3, 68, 9, 30, 3, 40, 9, 10

Offset

1,2

Links

Michel Marcus, Table of n, a(n) for n = 1..2000

Formula

a(n) + A343722(n) = n. - Michel Marcus, May 02 2021

Example

For every n >= 1, the residue r = 0 is a fixed point under the mapping r -> r^2 mod n, since we have 0 -> 0^2 mod n = 0. Also, for every n >= 2, the residue r = 1 is a fixed point, since we have 1 -> 1^2 mod n = 1.
For n=1, the only residue mod n is 0 (a fixed point), so a(1) = 1.
For n=2, the only residues are 0 and 1 (each a fixed point), so a(2) = 2.
For n=3, the only residue other than 0 and 1 is 2; 2 -> 2^2 mod 3 = 4 mod 3 = 1, a fixed point, so a(3) = 3.
For n=4, we have 0 -> 0, 1 -> 1, 2 -> 2^2 mod 4 = 4 mod 4 = 0, and 3 -> 3^2 mod 4 = 9 mod 4 = 1, each trajectory ending at a fixed point, so a(4) = 4.
For n=5, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 1 -> 1
  3 -> 4 -> 1 -> 1
  4 -> 1 -> 1
(each ending at a fixed point), so a(5) = 5.
For n=6, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 4
  3 -> 3
  4 -> 4
  5 -> 1 -> 1
(each ending at a fixed point), so a(6) = 6.
For n=7, however, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 2 -> ...       (a loop)
  3 -> 2 -> 4 -> 2 -> ...  (a loop)
  4 -> 2 -> 4 -> ...       (a loop)
  5 -> 4 -> 2 -> 4 -> ...  (a loop)
  6 -> 1 -> 1
so only 3 of the 7 trajectories reach a fixed point, so a(7)=3.

Prog

(PARI) pos(list, r) = forstep (k=#list, 1, -1, if (list[k] == r, return (#list - k + 1)); );
isok(r, n) = {my(list = List()); listput(list, r); for (k=1, oo, r = lift(Mod(r, n)^2); my(i = pos(list, r)); if (i==1, return (1)); if (i>1, return(0)); listput(list, r); ); } \\ reaches a fixed point
a(n) = sum(r=0, n-1, isok(r, n)); \\ Michel Marcus, May 02 2021

Crossrefs

Cf. A277847, A343720, A343722.

Sequence in context: A320004 A319704 A331304 * A070675 A096894 A097751

Adjacent sequences: A343718 A343719 A343720 * A343722 A343723 A343724

Keyword

nonn

Author

Jon E. Schoenfield, Apr 26 2021

Status

approved