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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.
2
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
1,1
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.
EXAMPLE
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
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
PROG
(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)]
CROSSREFS
Sequence in context: A027150 A020141 A049478 * A163776 A050558 A331192
KEYWORD
nonn
AUTHOR
Christian Perfect, Oct 19 2015
STATUS
approved