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
KEYWORD
nonn
AUTHOR
Christian Perfect, Oct 19 2015
STATUS
approved