A263458 Deal a pack of n cards into two piles and gather them up, n/2 times. All n such that this reverses the order of the deck.

4, 6, 12, 22, 28, 30, 36, 46, 52, 60, 70, 78, 100, 102, 108, 126, 148, 150, 156, 166, 172, 180, 190, 196, 198, 222, 228, 238, 262, 268, 270, 276, 292, 310, 316, 348, 358, 366, 372, 382, 388, 396, 420, 430, 438, 460, 462, 478, 486, 502, 508, 540, 556, 598

Offset: 1

Author

Christian Perfect, Oct 19 2015

Keywords

• nonn

Comments

This seems to be A003628(n)-1; that is, each element of this sequence is one less than a prime congruent to 5 or 7 modulo 8.

Examples

			Take a deck of 52 playing cards. Deal it into two piles, then pick up the first pile and put it on top of the other. Do this 26 times. The order of the deck is reversed, so 52 belongs to this sequence.
6 is in the sequence because the 3 shuffles are [1, 2, 3, 4, 5, 6] -> [5, 3, 1, 6, 4, 2] -> [4, 1, 5, 2, 6, 3] -> [6, 5, 4, 3, 2, 1], original reversed. 8 is not in the sequence because the 4 shuffles are [1, 2, 3, 4, 5, 6, 7, 8] -> [7, 5, 3, 1, 8, 6, 4, 2] -> [4, 8, 3, 7, 2, 6, 1, 5] -> [1, 2, 3, 4, 5, 6, 7, 8] -> [7, 5, 3, 1, 8, 6, 4, 2], not the original reversed. - _R. J. Mathar_, Aug 02 2024
		

Crossrefs

Cf. A003628, A373416.

Programs

Maple

isA263458 := proc(n)
    local L,itr ;
    L := [seq(i,i=1..n)] ;
    for itr from 1 to n/2 do
        L := pileShuf(L) ; # function code in A323712
    end do:
    for i from 1 to nops(L) do
        if op(-i,L) <> i then
            return false ;
        end if;
    end do:
    true ;
end proc:
n := 1;
for k from 2 do
    if isA263458(k) then
        printf("%d %d\n",n,k) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 02 2024

Sage

from itertools import cycle
def into_piles(r,deck):
    packs = [[] for i in range(r)]
    for card, pack in zip(range(1,deck+1),cycle(range(r))):
        packs[pack].insert(0,card)
    out = sum(packs,[])
    return Permutation(out)
def has_reversing_property(deck):
    p = power(into_piles(2,deck), deck/2)
    return p==into_piles(1,deck)
[i for i in range(2,400,2) if has_reversing_property(i)]