A217948 -id:A217948 - cp's OEIS frontend

A051733 Numbers n such that A051732(n) = n-1.

2, 3, 6, 7, 10, 15, 19, 27, 30, 31, 34, 42, 51, 54, 66, 70, 75, 82, 87, 90, 91, 99, 106, 114, 135, 147, 159, 174, 175, 187, 190, 195, 210, 211, 222, 231, 234, 246, 255, 262, 271, 274, 279, 282, 294, 307, 310, 327, 330, 331, 339, 351, 355, 379, 387, 394, 399, 411, 414

Offset: 1

Marie-Christine Haton (Marie-Christine.Haton(AT)loria.fr)

Olivier Gérard and Vincenzo Librandi, Table of n, a(n) for n = 1..6000 (first 386 terms from Olivier Gérard).

Cf. A051732. Equals A217948(n)/2.

(* v8 *) Select[Range[2,4000],Function[n,Sort[First[First[PermutationCycles@Join[Table[2r-1,{r,1,n}],Table[2r-2n,{r,n+1,2n}]]]]]==Range[2,2n-1]]] (* Olivier Gérard, Nov 08 2012 *)

A318712 Numbers k for which a riffle shuffle permutation with two or more packets permutes all except the first and the last of the k cards.

Original entry on oeis.org

4, 6, 12, 14, 18, 20, 30, 38, 42, 54, 60, 62, 68, 84, 90, 98, 102, 108, 110, 114, 132, 138, 140, 150, 164, 174, 180, 182, 198, 212, 228, 230, 234, 252, 258, 270, 282, 294, 308, 318, 348, 350, 354, 374, 380, 390, 402, 420, 422, 434, 444, 450, 462, 468, 492, 500, 510, 522, 524, 542, 548, 558, 564

Offset: 1

Mauro Rigo, Sep 01 2018

nonn

If k cards are cut into m packets of s cards each, the r-th card of the deck can be identified with the pair (p,q), where p=(r-1)%s+1 ('%' is the Euclidean division) is the number of the packet and q=((r-1) mod s)+1 is the position in the packet; e.g., k=4, m=2, s=2: 1->(1,1), 2->(1,2), 3->(2,1), 4->(2,2).
The shuffle is performed by taking the card with the lowest q of each packet in sequence and stacking them on top of one another. It is a generalization of the riffle shuffle with two cards (cf. A217948) and it can be described as a permutation where (p,q) becomes (((q-1)*m+p-1)%s+1,(((q-1)*m+p-1) mod s)+1). For example, k=8, m=4, s=2, (1,1)->(1,1), (2,1)->(1,2), (3,1)->(2,1), (4,1)->(2,2), (1,2)->(3,1), (2,2)->(3,2), (3,2)->(4,1), (4,2)->(4,2); this permutation can be described using the index r as (3,2,5)(4,6,7).
The terms of the sequence are the values of k for which the permutation consists of only one orbit with k-2 passages (e.g., k=6, m=3, s=2 is (3,2,4,5)).
Sequence A217948 is a subsequence of this one as it takes into consideration only the m=2 case.
If the formula is true then Tiago Januario's conjecture on A217948 would be solved (see formula).

Mauro Rigo, Table of n, a(n) for n = 1..5000

Cf. A217948, A225184.

arr = []; for i = 4:1000 pdiv = 2:ceil(sqrt(i)); divisors = pdiv(rem(i, pdiv)==0); stop = 0; for j = divisors if ~stop ndiv = j; neldiv = i/j; a0 = 1; b0 = 2; a = a0; b = b0; a1 = 0; b1 = 0; operations = 0; while a1~=a0 || b1~=b0 b1 = mod((b-1)*ndiv+a-1, neldiv)+1; a1 = floor(((b-1)*ndiv+a-1)/neldiv)+1; operations = operations + 1; a = a1; b = b1; end if operations==i-2 arr = [arr, i]; stop = 1; end end end end

Apparently a(n) = A225184(n+1) + 1.

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A051733 Numbers n such that A051732(n) = n-1.

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