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A323523
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Number of positive integer square matrices with entries summing to n and equal row and column sums.
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3
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1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 12, 1, 7, 22, 9, 1, 64, 1, 34, 121, 11, 1, 525, 2, 13, 407, 2022, 1, 801, 1, 10163, 1036, 17, 6211, 41735, 1, 19, 2212, 285784, 1, 3822, 1, 381446, 2229142, 23, 1, 1189540, 2, 22069276, 7261, 2309410, 1, 20943183, 164176641
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OFFSET
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0,5
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COMMENTS
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Also the number of non-normal semi-magic squares with positive integer entries summing to n.
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LINKS
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FORMULA
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EXAMPLE
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The a(12) = 12 matrices:
[12]
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[1 5] [5 1] [2 4] [4 2] [3 3]
[5 1] [1 5] [4 2] [2 4] [3 3]
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[1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
[1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
[2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 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]]}]];
ptnsqrs[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), And[SameQ@@Length/@#, Length[#]==0||Length[#]==Length[First[#]]]&];
Table[Sum[Length[Select[ptnsqrs[Times@@Prime/@y], And[SameQ@@Total/@#, SameQ@@Total/@Transpose[#]]&]], {y, IntegerPartitions[n]}], {n, 10}]
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CROSSREFS
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Cf. A000290, A006052, A007016, A103198, A120732, A257493, A321719, A321722, A323349, A323523, A323524, A323529.
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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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