A307551 Number of iterations of the map of quadratic residues x -> x^2 (mod prime(n)) with the initial term x = n^2 (mod prime(n)) needed to reach the end of the cycle.

0, 0, 1, 1, 3, 2, 3, 1, 9, 3, 3, 5, 5, 5, 10, 11, 27, 4, 9, 2, 3, 11, 19, 11, 4, 20, 7, 51, 17, 2, 5, 11, 9, 10, 35, 19, 11, 5, 81, 13, 10, 3, 35, 6, 21, 29, 11, 35, 27, 18, 27, 7, 5, 99, 7, 129, 65, 35, 10, 2, 22, 9, 23, 19, 13, 38, 19, 8, 171, 27, 13, 177, 59

Offset

1,5

Comments

Let L(n) be the number of elements in row n of A307550. Then a(n) = L(n) - 1.

Links

Table of n, a(n) for n=1..73.

Example

a(5) = 3 because prime(5) = 11, and 5^2 (mod 11) = 3 -> 3^2 (mod 11) = 9 ->  9^2 (mod 11) = 4 -> 4^2 (mod 11) = 5 with 3 iterations, where 5 is the last term of the cycle.

Maple

nn:=100:T:=array(1..3000):j:=0 :
for n from 1 to nn do:
p:=ithprime(n):lst0:={}:lst1:={}:ii:=0:r:=n:
for k from 1 to 10^6 while(ii=0) do:
  r1:=irem(r^2, p):lst0:=lst0 union {r1}:j:=j+1:T[j]:=r1:
      if lst0=lst1
       then
        ii:=1: printf(`%d, `, nops(lst0)-1):
        else
        r:=r1:lst1:=lst0:
      fi:
     od:
   if lst0 intersect {r1} = {r1}
    then
    j:=j-1:else fi:
od:

Mathematica

a[n_] := Module[{p = Prime[n]}, f[x_] := Mod[x^2, p]; Length[NestWhileList[f, f[n], Unequal, All]] - 2]; Array[a, 100] (* Amiram Eldar, Jul 05 2019 *)

Crossrefs

Cf. A000040, A000224, A046071, A063987, A096008, A307550.

Sequence in context: A197475 A195381 A144558 * A220344 A176102 A344220

Adjacent sequences: A307548 A307549 A307550 * A307552 A307553 A307554

Keyword

nonn

Author

Michel Lagneau, Apr 14 2019

Status

approved