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A323866
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Number of aperiodic toroidal necklaces of positive integers summing to n.
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10
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1, 1, 1, 3, 5, 12, 18, 42, 72, 145, 262, 522, 960, 1879, 3531, 6831, 13013, 25148, 48177, 93186, 179507, 347509, 671955, 1303257, 2527162, 4910681, 9545176, 18579471, 36183505, 70540861, 137603801, 268655547, 524842088, 1026067205, 2007118657, 3928564113
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OFFSET
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0,4
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COMMENTS
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The 1-dimensional (Lyndon word) case is A059966.
We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.
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LINKS
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EXAMPLE
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Inequivalent representatives of the a(6) = 18 toroidal necklaces:
[6] [1 5] [2 4] [1 1 4] [1 2 3] [1 3 2] [1 1 1 3] [1 1 2 2] [1 1 1 1 2]
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[1] [2] [1 1]
[5] [4] [1 3]
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[1] [1] [1]
[1] [2] [3]
[4] [3] [2]
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[1] [1]
[1] [1]
[1] [2]
[3] [2]
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[1]
[1]
[1]
[1]
[2]
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MATHEMATICA
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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]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m, {i, j}], {i, Length[m]}, {j, Length[First[m]]}]];
Table[If[n==0, 1, Length[Union@@Table[Select[ptnmats[k], And[apermatQ[#], neckmatQ[#]]&], {k, Times@@Prime/@#&/@IntegerPartitions[n]}]]], {n, 0, 10}]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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