

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.


1



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
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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:=j1: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 A318056
Adjacent sequences: A307548 A307549 A307550 * A307552 A307553 A307554


KEYWORD

nonn


AUTHOR

Michel Lagneau, Apr 14 2019


STATUS

approved



