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Andrew Howroyd, <a href="/A323867/b323867.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

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allocated for Gus WisemanNumber 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

0,4

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.

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

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}]

Cf. A000670, A000740, A027375, A101509.

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

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Gus Wiseman, Feb 04 2019

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