A307551 - OEIS

%I #11 Jul 08 2019 06:29:45

%S 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,

%T 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,

%U 129,65,35,10,2,22,9,23,19,13,38,19,8,171,27,13,177,59

%N 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.

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

%e 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.

%p nn:=100:T:=array(1..3000):j:=0 :

%p for n from 1 to nn do:

%p p:=ithprime(n):lst0:={}:lst1:={}:ii:=0:r:=n:

%p for k from 1 to 10^6 while(ii=0) do:

%p   r1:=irem(r^2,p):lst0:=lst0 union {r1}:j:=j+1:T[j]:=r1:

%p       if lst0=lst1

%p        then

%p         ii:=1: printf(`%d, `,nops(lst0)-1):

%p         else

%p         r:=r1:lst1:=lst0:

%p       fi:

%p      od:

%p    if lst0 intersect {r1} = {r1}

%p     then

%p     j:=j-1:else fi:

%p od:

%t 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 *)

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

%K nonn

%O 1,5

%A _Michel Lagneau_, Apr 14 2019