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
Keywords
Examples
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.
Programs
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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 *)
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