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A323867
Number of aperiodic arrays of positive integers summing to n.
12
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019
STATUS
approved