A225759 -id:A225759 - 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.

A225913 Composite numbers coprime to 6 such that A179382(n) = A000265(n-1), the odd part of n-1.

Original entry on oeis.org

517, 1525, 1765, 1837, 2941, 3241, 3265, 3397, 3421, 3565, 4117, 4501, 4669, 6349, 7357, 8569, 8749, 8965, 9085, 9997, 10045, 10057, 10981, 11929, 13741, 14101, 15757, 16117, 18745, 18949, 18997, 19885, 20557, 20629, 21805, 22045, 22645, 24445, 24505, 25417, 25429, 25681, 25885, 25957, 26101, 26245, 27205

Offset: 1

M. F. Hasler, May 20 2013

nonn

Inspired by the Conjecture mentioned in A225759.

forstep(n=1,9e9,6, A179382(n)==(n-1)>>valuation(n-1,2) & !isprime(n) & print1(n","))

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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