%I #13 Aug 23 2019 11:23:33

%S 1,2,4,12,32,60,216,252,912,1494,3960,4092,23904,16380,65016,130920,

%T 324960,262140,1569132,1048572,6281280,8388072,16769016,16777212,

%U 134150880,100663050,268402680,536865840,1610449344,1073741820,8589664080,4294967292,25768888320

%N Number of aperiodic binary arrays of size n.

%C An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

%H Andrew Howroyd, <a href="/A323864/b323864.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{d|n} A323860(d, n/d). - _Andrew Howroyd_, Aug 21 2019

%e The a(4) = 32 arrays:

%e [0001][0010][0011][0100][0110][0111][1000][1001][1011][1100][1101][1110]

%e .

%e [00] [00] [01] [01] [10] [10] [11] [11]

%e [01] [10] [00] [11] [00] [11] [01] [10]

%e .

%e [0] [0] [0] [0] [0] [0] [1] [1] [1] [1] [1] [1]

%e [0] [0] [0] [1] [1] [1] [0] [0] [0] [1] [1] [1]

%e [0] [1] [1] [0] [1] [1] [0] [0] [1] [0] [0] [1]

%e [1] [0] [1] [0] [0] [1] [0] [1] [1] [0] [1] [0]

%t apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];

%t zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);

%t Table[Length[Select[zaz[n],apermatQ]],{n,10}]

%Y Cf. A000740, A027375, A265627, A323351.

%Y Cf. A323860, A323862, A323863, A323865, A323867, A323869.

%K nonn

%O 0,2

%A _Gus Wiseman_, Feb 04 2019

%E Terms a(18) and beyond from _Andrew Howroyd_, Aug 21 2019