A163778 -id:A163778 - cp's OEIS frontend

A054639 Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, ..., n} -> {n, 1, n-1, 2, n-2, 3, ...} is of order n.

1, 2, 3, 5, 6, 9, 11, 14, 18, 23, 26, 29, 30, 33, 35, 39, 41, 50, 51, 53, 65, 69, 74, 81, 83, 86, 89, 90, 95, 98, 99, 105, 113, 119, 131, 134, 135, 146, 155, 158, 173, 174, 179, 183, 186, 189, 191, 194, 209, 210, 221, 230, 231, 233, 239

Offset: 1

Gilles Esposito-Farese (gef(AT)cpt.univ-mrs.fr), May 17 2000

nonn

The troubadour Arnaut Daniel composed sestinas based on the permutation 123456 -> 615243, which cycles after 6 iterations.
Roubaud quotes the number 141, but the corresponding Queneau-Daniel permutation is only of order 47 = 141/3.
This appears to coincide with the numbers n such that a type-2 optimal normal basis exists for GF(2^n) over GF(2). But are these two sequences really the same? - Joerg Arndt, Feb 11 2008
The answer is Yes - see Theorem 2 of the Dumas reference. [Jean-Guillaume Dumas (Jean-Guillaume.Dumas(AT)imag.fr), Mar 20 2008]
From Peter R. J. Asveld, Aug 17 2009: (Start)
a(n) is the n-th T-prime (Twist prime). For N >= 2, the family of twist permutations is defined by
p(m,N) == +2m (mod 2N+1) if 1 <= m < k = ceiling((N+1)/2),
p(m,N) == -2m (mod 2N+1) if k <= m < N.
N is T-prime if p(m,N) consists of a single cycle of length N.
The twist permutation is the inverse of the Queneau-Daniel permutation.
N is T-prime iff p=2N+1 is a prime number and exactly one of the following three conditions holds;
(1) N == 1 (mod 4) and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not,
(2) N == 2 (mod 4) and both +2 and -2 generate Z_p^*,
(3) N == 3 (mod 4) and -2 generate Z_p^* but +2 does not. (End)
The sequence name says the permutation is of order n, but P. R. J. Asveld's comment says it's an n-cycle. Is there a proof that those conditions are equivalent for the Queneau-Daniel permutation? (They are not equivalent for any arbitrary permutation; e.g., (123)(45)(6) has order 6 but isn't a 6-cycle.) More generally, I have found that for all n <= 9450, (order of Queneau-Daniel permutation) = (length of orbit of 1) = A003558(n). Does this hold for all n? - David Wasserman, Aug 30 2011

			For N=6 and N=7 we obtain the permutations (1 2 4 5 3 6) and (1 2 4 7)(3 6)(5): 6 is T-prime, but 7 is not. - _Peter R. J. Asveld_, Aug 17 2009

Raymond Queneau, Note complémentaire sur la Sextaine, Subsidia Pataphysica 1 (1963), pp. 79-80.
Jacques Roubaud, Bibliothèque Oulipienne No 65 (1992) and 66 (1993).

P. R. J. Asveld, Table of n, a(n) for n = 1..10085
Jean-Paul Allouche, Manon Stipulanti, and Jia-Yan Yao, Doubling modulo odd integers, generalizations, and unexpected occurrences, arXiv:2504.17564 [math.NT], 2025.
Joerg Arndt, Matters Computational (The Fxtbook), section 42.9 "Gaussian normal bases", pp. 914-920
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers, TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some families of permutations and their primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Queneau Numbers--Recent Results and a Bibliography, University of Twente, 2013.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.
Michèle Audin, Poésie, Spirales, et Battements de Cartes, Images des Mathématiques, CNRS, 2019 (in French).
M. Bringer, Sur un problème de R. Queneau, Math. Sci. Humaines No. 25 (1969) 13-20.
Jean-Guillaume Dumas, Caractérisation des Quenines et leur représentation spirale, Mathématiques et Sciences Humaines, Centre de Mathématique Sociale et de statistique, EPHE, 2008, 184 (4), pp. 9-23, hal-00188240.
G. Esposito-Farese, C program
Index entries for sequences related to the Josephus Problem

Not to be confused with Queneau's "s-additive sequences", see A003044.
A005384 is a subsequence.
Union of A163782 (Josephus_2-primes) and A163781 (dual Josephus_2-primes); also the union of A163777 (Archimedes_0-primes) and A163778 (Archimedes_1-primes); also the union of A071642/2 (shuffle primes) and A163776/2 (dual shuffle primes). - Peter R. J. Asveld, Aug 17 2009
Cf. A216371, A003558 (for which a(n) == n).

QD:= proc(n) local i;
  if n::even then map(op,[seq([n-i,i+1],i=0..n/2-1)])
  else map(op, [seq([n-i,i+1],i=0..(n-1)/2-1),[(n+1)/2]])
  fi
end proc:
select(n -> GroupTheory:-PermOrder(Perm(QD(n)))=n, [$1..1000]); # Robert Israel, May 01 2016

a[p_] := Sum[Cos[2^n Pi/((2 p + 1) )], {n, 1, p}];
Select[Range[500],Reduce[a[#] == -1/2, Rationals] &] (* Gerry Martens, May 01 2016 *)

is(n)=
{
    if (n==1, return(1));
    my( m=n%4 );
    if ( m==4, return(0) );
    my(p=2*n+1, r=znorder(Mod(2,p)));
    if ( !isprime(p), return(0) );
    if ( m==3 && r==n, return(1) );
    if ( r==2*n, return(1) ); \\ r == 1 or 2
    return(0);
}
for(n=1,10^3, if(is(n),print1(n,", ")) );
\\ Joerg Arndt, May 02 2016

a(n) = (A216371(n)-1)/2. - L. Edson Jeffery, Dec 18 2012

a(n) >> n log n, and on the Bateman-Horn-Stemmler conjecture a(n) << n log^2 n. I imagine a(n) ≍ n log n, and numerics suggest that perhaps a(n) ~ kn log n for some constant k (which seems to be around 1.122). - Charles R Greathouse IV, Aug 02 2023

A163777 Even terms in the sequence of Queneau numbers A054639.

Original entry on oeis.org

2, 6, 14, 18, 26, 30, 50, 74, 86, 90, 98, 134, 146, 158, 174, 186, 194, 210, 230, 254, 270, 278, 306, 326, 330, 338, 350, 354, 378, 386, 398, 410, 414, 426, 438, 470, 530, 554, 558, 606, 614, 618, 638, 650, 686, 690, 726, 746, 774, 810, 818, 834, 846, 866, 870

Offset: 1

Peter R. J. Asveld, Aug 11 2009

nonn

Previous name was: a(n) is the n-th A_0-prime (Archimedes_0 prime).

We have: (1) N is A_0-prime if and only if N is even, p = 2N + 1 is a prime number and both +2 and -2 generate Z_p^* (the multiplicative group of Z_p); (2) N is A_0-prime if and only if N = 2 (mod 4), p = 2N + 1 is a prime number and both +2 and -2 generate Z_p^*.

P. R. J. Asveld, Table of n, a(n) for n = 1..3378.
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011), 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. Another link.
Index entries for sequences related to the Josephus Problem

The A_0-primes are the even T- or Twist-primes, these T-primes are equal to the Queneau-numbers (A054639). For the related A_1-, A^+_1- and A^-_1-primes, see A163778, A163779 and A163780. Considered as sets A163777 is the intersection of the Josephus_2-primes (A163782) and the dual Josephus_2-primes (A163781), it also equals the difference of A054639 and the A_1-primes (A163779).

Cf. A137310.

okQ[n_] := EvenQ[n] && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* Jean-François Alcover, Sep 10 2019, from PARI *)

Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
ok(n)={n>1 && n==Follow(1, j->ceil((n+1)/2) - (-1)^j*ceil((j-1)/2))}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

ok(n)={n%2==0 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == 2*n}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(n) = 2*A137310(n). - Andrew Howroyd, Nov 11 2017

Definition simplified by Michel Marcus, May 27 2013
a(33)-a(55) from Andrew Howroyd, Nov 11 2017
New name from Joerg Arndt, Mar 23 2018, edited by M. F. Hasler, Mar 24 2018

A163780 Terms in A054639 equal to 3 mod 4.

Original entry on oeis.org

3, 11, 23, 35, 39, 51, 83, 95, 99, 119, 131, 135, 155, 179, 183, 191, 231, 239, 243, 251, 299, 303, 323, 359, 371, 375, 411, 419, 431, 443, 483, 491, 495, 515, 519, 531, 543, 575, 611, 615, 639, 651, 659, 683, 719, 723, 743, 755, 771, 779, 783, 791, 803, 831, 879

Offset: 1

Peter R. J. Asveld, Aug 12 2009

nonn

Previous name was: a(n) is the n-th A^-_1-prime (Archimedes^-_1 prime).

N is A^-_1-prime iff N=3 (mod 4), p=2N+1 is a prime number and -2 generates Z_p^* (the multiplicative group of Z_p) but +2 does not.

P. R. J. Asveld Table of n, a(n) for n=1..3378.
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.

The A^-_1-primes are the T- or Twist-primes congruent 3 (mod 4), these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A_1- and A^+_1-primes, see A163777, A163778 and A163779. Considered as sets the union of A163779 and A163780 equals A163778, the union of A163780 and A163777 is equal to A163781 (dual J_2-primes).

ok(n) = n%4==3 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == n;
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(33)-a(55) from Andrew Howroyd, Nov 11 2017

New name from Andrey Zabolotskiy, Mar 23 2018

A105876 Primes for which -4 is a primitive root.

Original entry on oeis.org

3, 7, 11, 19, 23, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 163, 167, 179, 191, 199, 211, 227, 239, 263, 271, 311, 347, 359, 367, 379, 383, 419, 443, 463, 467, 479, 487, 491, 503, 523, 547, 563, 587, 599, 607, 619, 647, 659, 719, 743, 751, 787, 823, 827, 839, 859, 863

Offset: 1

N. J. A. Sloane, Apr 24 2005

nonn

Also, primes for which -16 is a primitive root. For proof see following comments from Michael Somos, Aug 07 2009:
Let p = 8*t + 3 be prime. It is well-known that 2 is a primitive root.
We will use the obvious fact that if a primitive root is a power of another element, then that other element is also a primitive root. So
-1 == 2^(4*t+1) (mod p) because 2 is primitive root.
-2 == 2^(4*t+2) == 4^(2*t+1) (mod p) obvious
2 == (-4)^(2*t+1) (mod p) obvious, therefore -4 is also primitive root.
2 == 2^(8*t+3) (mod p) obviously works not just for 2
4 == 2^(8*t+4) == 16^(2*t+1) (mod p) obvious
-4 == (-16)^(2*t+1) (mod p) obvious, therefore -16 is also primitive root.
The case where p = 8*t + 7 is similar.
From Jianing Song, Dec 24 2022: (Start)
Equivalently, primes p == 3 (mod 4) such that the multiplicative order of 4 modulo p is (p-1)/2 (a subsequence of A216371).
Proof of equivalence: let ord(a,k) be the multiplicative of a modulo k. First we notice that all terms are congruent to 3 modulo 4, since -4 is a quadratic residue modulo p if p == 1 (mod 4). If ord(4,p) = (p-1)/2. Note that (p-1)/2 is odd, so it is coprime to ord(-1,p) = 2. As a result, ord(-4,p) = ((p-1)/2) * 2 = p-1. Conversely, If ord(-4,p) = p-1, we must have ord(4,p) = (p-1)/2 by noting that ord(-4,p) <= lcm(2,ord(4,p)).
Also primes p such that the multiplicative order of 16 modulo p is (p-1)/2. Proof: note that ord(16,p) = ord(-4,p)/gcd(ord(-4,p),2). If ord(-4,p) = p-1, then ord(16,p) = (p-1)/2. Conversely, if ord(16,p) = (p-1)/2, then ord(-4,p) = p-1, since otherwise ord(-4,p) = (p-1)/2 is odd, which is impossible since that -4 is not a quadratic residue modulo a prime p == 3 (mod 4).
{(a(n)-1)/2} is the sequence of indices of fixed points of A302141.
An odd prime p is a term if and only if p == 3 (mod 4) and the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 if p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)).
It seems that a(n) = 2*A163778(n-1) + 1 for n >= 2. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for primes by primitive root

Cf. A114564, A302141, A163778. A216371 is a supersequence.

pr=-4; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* OR *)
a[p_, q_]:=Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))],{n,1,2 q p}]
2 Select[Range[500],Rationalize[N[a[#,2],20]]==1 &]+1
(* Gerry Martens, Apr 28 2015 *)

is(n)=isprime(n) && n>2 && znorder(Mod(-4,n))==n-1 \\ Charles R Greathouse IV, Apr 30 2015

Edited by N. J. A. Sloane, Aug 08 2009

A163779 Numbers k of the form 4j + 1 such that 2k + 1 is a prime with primitive root 2.

Original entry on oeis.org

1, 5, 9, 29, 33, 41, 53, 65, 69, 81, 89, 105, 113, 173, 189, 209, 221, 233, 245, 261, 273, 281, 293, 309, 329, 393, 413, 429, 441, 453, 473, 509, 545, 561, 585, 593, 629, 641, 645, 653, 713, 725, 741, 749, 761, 765, 785, 809, 833, 873, 893, 933, 953, 965, 989, 993

Offset: 1

Peter R. J. Asveld, Aug 12 2009

nonn

Previous name was: a(n) is the n-th A^+_1-prime (Archimedes^+_1 prime).

N is A^+_1-prime iff N=1 (mod 4), p=2N+1 is a prime number and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not.

Joerg Arndt, Table of n, a(n) for n = 1..10000 (first 3328 from P. R. J. Asveld)
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.
P. Michael Hutchins, Reworded Definition

The A^+_1-primes are the T- or Twist-primes congruent 1 (mod 4), these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A_1- and A^-_1-primes, see A163777, A163778 and A163780. Considered as sets the union of A163779 and A163780 equals A163778, the union of A163779 and A163777 is equal to A163782 (J_2-primes).

okQ[n_] := Mod[n, 4] == 1 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* Jean-François Alcover, Jun 30 2018, after Andrew Howroyd *)

ok(n) = n%4==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1))==2*n;
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

2 * a(n) + 1 = A213051(n+1). - Joerg Arndt, Mar 23 2018

a(32)-a(55) from Andrew Howroyd, Nov 11 2017

Term 1 prepended and new name from Joerg Arndt, Mar 23 2018

A294673 Order of the "inside-out" permutation on 2n+1 letters.

Original entry on oeis.org

1, 3, 5, 4, 9, 11, 9, 5, 12, 12, 7, 23, 8, 20, 29, 6, 33, 35, 20, 39, 41, 28, 12, 36, 15, 51, 53, 36, 44, 24, 20, 7, 65, 36, 69, 60, 42, 15, 20, 52, 81, 83, 9, 60, 89, 60, 40, 95, 12, 99, 84, 66, 105, 28, 18, 37, 113, 30, 92, 119, 81, 36, 25, 8, 36, 131, 22, 135, 20, 30, 47, 60, 48, 116, 132, 100, 51, 155

Offset: 0

P. Michael Hutchins, Nov 06 2017

The "inside-out" permutation (closely related to the Mongean shuffle, see A019567) sends (t_1, t_2, ..., t_{2n+1}) to (t_{n+1}, t_{n+2}, t_{n}, t_{n+3}, t_{n-1}, ..., t_1). For n = 0, 1, 2, 3, this is (1), (2,3,1), (3,4,2,5,1), (4,5,3,6,2,7,1), whose orders are respectively 1,3,5,4.

This is the odd bisection of A238371 and also the odd bisection of A003558 (see Joseph L. Wetherell's comment below).

			For n=2: Iterating the "inside-out" permutation of a string of length 2n+1=5:
12345
34251
25413
41532
53124
12345
...
which has order a(2) = 5.

Robert Israel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, a(n) for n = 0..32683 (computed using Robert Israel's Maple program)

Cf. A003558, A019567, A163778, A238371, A294434.

f:=func;
[f(n): n in [0..100]]; // Joseph L. Wetherell, Nov 12 2017

[Order(Integers(4*n+3)!-2*(-1)^n): n in [0..100]]; // Joseph L. Wetherell, Nov 15 2017

f:= proc(n)
  ilcm(op(map(nops,convert(map(op, [[n+1],seq([n+1+i,n+1-i],i=1..n)]),disjcyc))))
end proc:
map(f, [$0..100]); # Robert Israel, Nov 09 2017

a[n_] := MultiplicativeOrder[-2(-1)^n, 4n+3];
a /@ Range[0, 100] (* Jean-François Alcover, Apr 07 2020 *)

Follow(s,f)={my(t=f(s),k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
CyclePoly(n,x)={my(p=0); for(i=1, 2*n+1, my(l=Follow(i, j->n+1+(-1)^j*ceil((j-1)/2) )); if(l,p+=x^l)); p}
a(n)={my(p=CyclePoly(n,x), m=1); for(i=1,poldegree(p),if(polcoeff(p,i), m=lcm(m,i))); m} \\ Andrew Howroyd, Nov 08 2017

a(n)=znorder(Mod(if(n%2,2,-2),4*n+3)) \\ See Wetherell formula; Charles R Greathouse IV, Nov 15 2017

The permutation sends i (1 <= i <= 2n+1) to p(i) = n + 1 + f(i), where f(i) = (-1)^i*ceiling((i-1)/2).
a(n) = minimal k>0 such that p^k() = p^0().
a((A163778(n)-1)/2) = A163778(n). - Andrew Howroyd, Nov 11 2017.
From Joseph L. Wetherell, Nov 14 2017: (Start)
a(n) is equal to the order of multiplication-by-2 acting on the set of nonzero elements in (Z/(4n+3)Z), modulo the action of +-1. To be precise, identify i=1,2,...,2*n+1 with the odd representatives J=1,3,...,4*n+1 of this set, via the map J = 2*i-1. It is not hard to show that the induced permutation on the set of J values is given on integer representatives by J -> (4*n+3+J)/2 if i=(J+1)/2 is even and J -> (4*n+3-J)/2 if i=(J+1)/2 is odd. It follows that this induces the permutation J -> +-J/2 (mod 4*n+3), from which we immediately see that the order is as stated.
Note that the order of 2 acting on (Z/(4n+3)Z)/{+-1} is the same as the order of either 2 or -2 acting on (Z/(4n+3)Z), depending on which of these is a quadratic residue modulo 4n+3. Thus an equivalent (and often easier) way to compute a(n) is as the order of -2*(-1)^n acting on (Z/(4n+3)Z).
Among other things, the lower and upper bounds log_2(n) + 2 < a(n) <= 2*n+1 follow immediately.
(End)
It appears that the upper bound a(n) = 2n+1 occurs iff 2n+1 belongs to A163778 or equivalently iff n belongs to A294434. This almost (but not quite) follows from the above comments by Andrew Howroyd and Joseph L. Wetherell. - N. J. A. Sloane, Nov 16 2017

A238371 a(1)=1; for n > 1, a(n) = the number of "topped" Mongean shuffles to reorder a stack of n cards to its original order.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 4, 4, 9, 9, 11, 11, 9, 9, 5, 5, 12, 12, 12, 12, 7, 7, 23, 23, 8, 8, 20, 20, 29, 29, 6, 6, 33, 33, 35, 35, 20, 20, 39, 39, 41, 41, 28, 28, 12, 12, 36, 36, 15, 15, 51, 51, 53, 53, 36, 36, 44, 44, 24, 24, 20, 20, 7, 7, 65, 65, 36, 36, 69, 69, 60, 60, 42, 42, 15, 15, 20, 20, 52, 52, 81, 81, 83, 83, 9, 9, 60, 60

Offset: 1

R. J. Mathar, Feb 25 2014

nonn

In the Mongean shuffle, the top card of the stack becomes the top of the new stack, the second of the old stack goes on top of the new stack, the third to the bottom of the new stack, alternating top and bottom of the new stack.
Here we define a shuffle where the top-bottom placements in the new stack alternate in the same way, but the second card of the old stack moves to the *bottom* of the stack.
A single shuffle is a permutation of 1, 2, 3, 4, 5, 6, ... -> ..., 7, 5, 3, 1, 2, 4, 6, ...
The fixed points, where n=a(n), seem to be in A163778.
(The "topped" classification is a nomenclature invented here, to be replaced if this variant appears elsewhere in the literature.)

Wikipedia, Mongean shuffle

Cf. A019567 (Mongean shuffle), A294673 (a bisection).

topMong := proc(L)
    ret := [op(1,L)] ;
    for k from 2 to nops(L) do
        if type(k,'even') then
            ret := [op(ret),op(k,L)] ;
        else
            ret := [op(k,L),op(ret)] ;
        end if;
    end do:
    ret ;
end proc:
A238371 := proc(n)
    local ca,org,tu ;
    ca := [seq(k,k=1..n)] ;
    org := [seq(k,k=1..n)] ;
    for tu from 1 do
        ca := topMong(ca) ;
        if ca = org then
            return tu;
        end if:
    end do:
end proc:
seq(A238371(n),n=2..88) ;

topMong[L_] := Module[{ret = {L[[1]]}}, For[k = 2, k <= Length[L], k++, If[ EvenQ[k], ret = Append[ret, L[[k]]], ret = Prepend[ret, L[[k]]]]]; ret];
A238371[n_] := Module[{ca, org, tu}, ca = org = Range[n]; For[tu = 1, True, tu++, ca = topMong[ca]; If[ca == org, Return[tu]]]];
Array[A238371, 88] (* Jean-François Alcover, Jul 03 2018, after R. J. Mathar *)

apply( A238371(n)=znorder(Mod(bitand(n,2)*2-2,n\2*4+3)), [0..99]) \\ M. F. Hasler, Mar 31 2019

a(A163778(n)) = A163778(n). - Andrew Howroyd, Nov 11 2017

A302141 Multiplicative order of 16 mod 2n+1.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 3, 1, 2, 9, 3, 11, 5, 9, 7, 5, 5, 3, 9, 3, 5, 7, 3, 23, 21, 2, 13, 5, 9, 29, 15, 3, 3, 33, 11, 35, 9, 5, 15, 39, 27, 41, 2, 7, 11, 3, 5, 9, 12, 15, 25, 51, 3, 53, 9, 9, 7, 11, 3, 6, 55, 5, 25, 7, 7, 65, 9, 9, 17, 69, 23, 15, 7, 21, 37, 15, 6, 5, 13, 13, 33, 81, 5, 83, 39, 9, 43, 15, 29, 89, 45, 15, 9, 10, 9, 95, 24, 3, 49, 99, 33

Offset: 0

Jianing Song, Apr 02 2018

nonn

Reptend length of 1/(2n+1) in hexadecimal.
a(n) <= n; it appears that equality holds if and only if n=1 or is in A163778. - Robert Israel, Apr 02 2018
From Jianing Song, Dec 24 2022: (Start)
a(n) <= psi(2*n+1)/2 <= n. a(n) = psi(2*n+1)/2 if and only if the multiplicative order of 2 modulo 2*n+1 is psi(2*n+1) or psi(2*n+1)/2, and psi(2*n+1) == 2 (mod 4).
a(n) = n if and only if A053447(n) = n and A053447(n) is odd. As a result, a(n) = n if and only if 2*n+1 = p is a prime congruent to 3 modulo 4, and the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 if p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)). Such primes p are listed in A105876. (End)

			The fraction 1/13 is equal to 0.13B13B... in hexadecimal, so a(6) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Jianing Song)
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A053447, A053451, A105876, A163778.

List([0..100],n->OrderMod(16,2*n+1)); # Muniru A Asiru, Feb 25 2019

[1] cat [ Modorder(16, 2*n+1): n in [1..100]]; // Vincenzo Librandi, Apr 03 2018

seq(numtheory:-order(16, 2*n+1), n=0..100); # Robert Israel, Apr 02 2018

Table[MultiplicativeOrder[16, 2 n + 1], {n, 0, 150}] (* Vincenzo Librandi, Apr 03 2018 *)

a(n) = znorder(Mod(16, 2*n+1)) \\ Felix Fröhlich, Apr 02 2018

a(n) = A002326(n)/gcd(A002326(n),4) = A053447(n)/gcd(A053447(n),2). [Corrected by Jianing Song, Dec 24 2022]

cp's OEIS Frontend

A294434 a(0)=0; for n>0, a(n) = (A163778(n)-1)/2.

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A054639 Queneau numbers: numbers n such that the Queneau-Daniel permutation {1, 2, 3, ..., n} -> {n, 1, n-1, 2, n-2, 3, ...} is of order n.

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A163777 Even terms in the sequence of Queneau numbers A054639.

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A163780 Terms in A054639 equal to 3 mod 4.

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A105876 Primes for which -4 is a primitive root.

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A163779 Numbers k of the form 4*j + 1 such that 2*k + 1 is a prime with primitive root 2.

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A294673 Order of the "inside-out" permutation on 2n+1 letters.

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A163779 Numbers k of the form 4j + 1 such that 2k + 1 is a prime with primitive root 2.