A001134 -id:A001134 - cp's OEIS frontend

A384184 Order of the permutation of {0,...,n-1} formed by successively swapping elements at i and 2*i mod n, for i = 0,...,n-1.

1, 2, 1, 4, 2, 2, 2, 8, 3, 4, 5, 4, 6, 4, 6, 16, 4, 6, 9, 8, 4, 10, 28, 8, 10, 12, 9, 8, 14, 12, 12, 32, 5, 8, 70, 12, 18, 18, 24, 16, 10, 8, 7, 20, 210, 56, 126, 16, 110, 20, 60, 24, 26, 18, 120, 16, 9, 28, 29, 24, 30, 24, 60, 64, 6, 10, 33, 16

Offset: 1

Mia Boudreau, May 29 2025

nonn

a(2*n) = 2*a(n) since the cycle lengths of the permutation with size 2*n is effectively that of size n twice, doubled. Thus, the LCM/order is doubled.

			For n = 11, the permutation is {0,3,4,7,8,1,2,9,10,5,6} and it has order a(11) = 5.

Mia Boudreau, Table of n, a(n) for n = 1..10000
Mia Boudreau, Java program, GitHub

Cf. A003418, A000027, A051732, A004626, A065119, A001122, A155072, A001133, A001134, A001135, A225759.

from sympy.combinatorics import Permutation
def a(n):
   L = list(range(n))
   for i in range(n):
       if (j:= (i << 1) % n) != i:
           L[i],L[j] = L[j],L[i]
   return Permutation(L).order() # Darío Clavijo, Jun 05 2025

a(2*n) = 2*a(n).
a(2^n) = 2^n.
Conjecture: a(2^n + 2^x) = 2^n * (x-n) if x > n.
a(2^n - 1) = A003418(n-1).
s(2^n + 1) = A000027(n).
a(2*n - 1) = A051732(n).
a(A004626(n)) % 2 = 1.
a(A065119(n)) = n/3.
a(A001122(n)) = (n-1) / 2.
a(A155072(n)) = (n-1) / 4.
a(A001133(n)) = (n-1) / 6.
a(A001134(n)) = (n-1) / 8.
a(A001135(n)) = (n-1) / 10.
a(A225759(n)) = (n-1) / 16.

A141230 Odd numbers n for which A006694((n-1)/2)=4.

Original entry on oeis.org

15, 33, 39, 49, 55, 57, 81, 87, 95, 111, 113, 143, 159, 177, 183, 201, 209, 249, 281, 289, 295, 303, 319, 321, 335, 353, 393, 407, 415, 417, 447, 489, 519, 529, 535, 537, 543, 551, 577, 583, 591, 593, 617, 625, 633, 649, 655, 681, 695, 737, 767, 807, 815, 879, 895, 913, 951

Offset: 1

Vladimir Shevelev, Jun 16 2008

nonn

If p>3 is a prime then 3p is in this sequence if and only if p is in A001122.

Cf. A006694, A001122, A001134.

a006694(n)=sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1;
isok(n) = (n % 2) && (a006694((n-1)/2)== 4); \\ Michel Marcus, Dec 18 2018

More terms from Michel Marcus, Dec 18 2018

cp's OEIS Frontend

A384184 Order of the permutation of {0,...,n-1} formed by successively swapping elements at i and 2*i mod n, for i = 0,...,n-1.

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