A294673 -id:A294673 - cp's OEIS frontend

A305151 a(n) = (2n+1) - A294673(n), the amount by which A294673 is less than the maximum possible for n.

0, 0, 0, 3, 0, 0, 4, 10, 5, 7, 14, 0, 17, 7, 0, 25, 0, 0, 17, 0, 0, 15, 33, 11, 34, 0, 0, 19, 13, 35, 41, 56, 0, 31, 0, 11, 31, 60, 57, 27, 0, 0, 76, 27, 0, 31, 53, 0, 85, 0, 17, 37, 0, 79, 91, 74, 0, 85, 25, 0, 40, 87, 100, 119, 93, 0, 111, 0, 117, 109, 94, 83, 97, 31, 17, 51, 102, 0

Offset: 0

P. Michael Hutchins, May 26 2018

nonn

Motivation: A294673(n), which is always <= 2n+1, equals n in most cases. This sequence abstracts out that commonality to leave a cleaner signal.

Cf. A294673.

a(n) = (2*n+1) - znorder(Mod(if(n%2, 2, -2), 4*n+3)); \\ Michel Marcus, May 26 2018

A163778 Odd terms in A054639.

Original entry on oeis.org

3, 5, 9, 11, 23, 29, 33, 35, 39, 41, 51, 53, 65, 69, 81, 83, 89, 95, 99, 105, 113, 119, 131, 135, 155, 173, 179, 183, 189, 191, 209, 221, 231, 233, 239, 243, 245, 251, 261, 273, 281, 293, 299, 303, 309, 323, 329, 359, 371, 375, 393, 411, 413, 419, 429

Offset: 1

Peter R. J. Asveld, Aug 11 2009

nonn

Previous name was: The A_1-primes (Archimedes_1 primes).
We have: (1) N is an A_1-prime iff N is odd, p=2N+1 is a prime number and only one of +2 and -2 generates Z_p^* (the multiplicative group of Z_p); (2) N is an A_1-prime iff p=2N+1 is a prime number and exactly one of the following holds: (a) N == 1 (mod 4) and +2 generates Z_p^* but -2 does not, (b) N == 3 (mod 4) and -2 generates Z_p^* but +2 does not.
The A_1-primes are the odd T- or Twist-primes (the T-primes are the same as the Queneau-numbers, A054639). For the related A_0-, A^+_1- and A^-_1-primes, see A163777, A163779 and A163780. Considered as a set, the present sequence is the union of the A^+_1-primes (A163779) and the A^-_1-primes (A163780). It is also equal to the difference of A054639 and the A_0-primes (A163777).

P. R. J. Asveld, Table of n, a(n) for n=1..6706.
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.

Cf. A054639, A163777, A163779, A163780, A294434, A294673.

follow[s_, f_] := Module[{t, k}, t = f[s]; k = 1; While[t>s, k++; t = f[t]]; If[s == t, k, 0]];
okQ[n_] := n>1 && n == follow[1, Function[j, Ceiling[n/2] + (-1)^j*Ceiling[ (j-1)/2]]];
A163778 = Select[Range[1000], okQ] (* Jean-François Alcover, Jun 07 2018, after Andrew Howroyd *)

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/2) + (-1)^j*ceil((j-1)/2))}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

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

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

New name from Joerg Arndt, Mar 23 2018

A019567 Order of the Mongean shuffle permutation of 2n cards: a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1.

Original entry on oeis.org

1, 2, 3, 6, 4, 6, 10, 14, 5, 18, 10, 12, 21, 26, 9, 30, 6, 22, 9, 30, 27, 8, 11, 10, 24, 50, 12, 18, 14, 12, 55, 50, 7, 18, 34, 46, 14, 74, 24, 26, 33, 20, 78, 86, 29, 90, 18, 18, 48, 98, 33, 10, 45, 70, 15, 24, 60, 38, 29, 78, 12, 84, 41, 110, 8, 84, 26, 134, 12, 46, 35, 36, 68, 146

Offset: 0

John Bullitt (metta(AT)world.std.com), N. J. A. Sloane and J. H. Conway

Write down 1, then 2 to left, 3 to right, 4 to left, ..., getting [ 2n,2n-2,...,4,2,1,3,5,...,2n-1 ]; the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives order of permutation sending 1 to 2n, 2 to 2n-2, ..., 2n to 2n-1.
Equivalently, the sequence 2,3,6,4,6,10,14,5,18,10,12,21,26,9,... gives the number of Mongean shuffles needed to return a deck of 2n cards (n=1,2,3,...) to its original order.
It appears that a(n) = order((-1)^(n+1)*2 in Z_{2n+1}) / f with f=1 when n==2 (mod 3) and for n = 0, 19, 21, 30,33, 52, 55, 61, 63, 70, ..., f=2 else. I don't know how to characterize the "exceptional" n's. - M. F. Hasler, Mar 31 2019

			Illustrating the initial terms:
   n  4n+1  2^m+1  2^m-1  m
   0    1            1    1
   1    5     5           2
   2    9     9           3
   3   13    5*13         6
   4   17     17          4
   5   21           3*21  6
   6   25   41*25        10

A. P. Domoryad, Mathematical Games and Pastimes, Pergamon Press, 1964; see pp. 134-135.
W. W. Rouse Ball, Mathematical Recreations and Essays, 11th ed. 1939, p. 311

R. J. Mathar, Table of n, a(n) for n = 0..2000
P. Diaconis, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196.
Arne Ledet, The Monge shuffle for two-power decks, Math. Scand. Vol 98, No 1 (2006), 5-11.
E. Ross, Mathematics and Music: The Mathieu Group M_12 (2011), Chapter 2.
T. & X. Vigouroux, First 2000000 terms, for n = 0..1999999

Cf. A163777, A238371, A294673.

A019567:=  proc(n)
    for m from 1 do
        if modp(2^m-1,4*n+1) =0 or modp(2^m+1,4*n+1)=0 then
            return m ;
        end if;
    end do;
end proc: # N. J. A. Sloane, Jul 28 2007

a[n_] := For[m=1, True, m++, If[AnyTrue[{-1, 1}, Divisible[2^m+#, 4n+1]&], Return[m]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 26 2019 *)

A019567(n,z=Mod(2,4*n+1))=for(m=1,oo,bittest(5,lift(z^m+1))&&return(m)) \\ M. F. Hasler, Mar 31 2019

a(A163777(n)/2) = A163777(n). - Andrew Howroyd, Nov 11 2017

Comments corrected by Mikko Nieminen, Jul 26 2007, who also provided the Domoryad reference

Definition edited by N. J. A. Sloane, Nov 09 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

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

Original entry on oeis.org

0, 1, 2, 4, 5, 11, 14, 16, 17, 19, 20, 25, 26, 32, 34, 40, 41, 44, 47, 49, 52, 56, 59, 65, 67, 77, 86, 89, 91, 94, 95, 104, 110, 115, 116, 119, 121, 122, 125, 130, 136, 140, 146, 149, 151, 154, 161, 164, 179, 185, 187, 196, 205, 206, 209, 214, 215, 220, 221

Offset: 0

N. J. A. Sloane, Nov 16 2017

nonn

Appears to be equal to the indices of maximal entries in A294673.

N. J. A. Sloane, Table of n, a(n) for n = 0..6706

Cf. A163778, A294673.

cp's OEIS Frontend

A305151 a(n) = (2n+1) - A294673(n), the amount by which A294673 is less than the maximum possible for n.

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A163778 Odd terms in A054639.

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A019567 Order of the Mongean shuffle permutation of 2n cards: a(n) is least number m for which either 2^m + 1 or 2^m - 1 is divisible by 4n + 1.

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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.

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

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