A323867 Number of aperiodic arrays of positive integers summing to n.

1, 1, 1, 5, 11, 33, 57, 157, 303, 683, 1358, 2974, 5932, 12560, 25328, 52400, 106256, 217875, 441278, 899955, 1822703, 3701401, 7491173, 15178253, 30691135, 62085846, 125435689, 253414326, 511547323, 1032427635, 2082551931, 4199956099, 8466869525, 17064777665

Offset

0,4

Comments

The 1-dimensional case is A000740.

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Example

The a(5) = 33 arrays:
  5  14  23  32  41  113  122  131  212  221  311  1112  1121  1211  2111
.
  1  2  3  4  11  11  12  21
  4  3  2  1  12  21  11  11
.
  1  1  1  2  2  3
  1  2  3  1  2  1
  3  2  1  2  1  1
.
  1  1  1  2
  1  1  2  1
  1  2  1  1
  2  1  1  1

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), SameQ@@Length/@#&];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}];
Table[Length[Union@@Table[Select[ptnmats[k], apermatQ], {k, Times@@Prime/@#&/@IntegerPartitions[n]}]], {n, 15}]

Prog

(GAP) List([0..30], A323867); # See A323861 for code; Andrew Howroyd, Aug 21 2019

Crossrefs

Cf. A000670, A000740, A027375, A101509.

Cf. A323860, A323861, A323862, A323863, A323864, A323866, A323869.

Sequence in context: A106908 A107442 A349611 * A280540 A077917 A127864

Adjacent sequences: A323864 A323865 A323866 * A323868 A323869 A323870

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019

Status

approved