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A323307
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Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.
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11
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1, 1, 2, 4, 2, 6, 3, 12, 18, 12, 2, 36, 4, 10, 20, 72, 2, 60, 4, 40, 60, 24, 3, 120, 80, 14, 360, 120, 4, 240, 2, 240, 42, 32, 70, 720, 6, 27, 112, 480, 2, 210, 4, 84, 420, 40, 4, 1440, 280, 280, 108, 224, 5, 1260, 224, 420, 180, 22, 2, 840, 6, 72, 1680, 2880
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OFFSET
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1,3
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COMMENTS
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This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
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LINKS
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FORMULA
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EXAMPLE
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The a(22) = 24 matrices:
[111112] [111121] [111211] [112111] [121111] [211111]
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[111] [111] [111] [112] [121] [211]
[112] [121] [211] [111] [111] [111]
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[11] [11] [11] [11] [12] [21]
[11] [11] [12] [21] [11] [11]
[12] [21] [11] [11] [11] [11]
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[1] [1] [1] [1] [1] [2]
[1] [1] [1] [1] [2] [1]
[1] [1] [1] [2] [1] [1]
[1] [1] [2] [1] [1] [1]
[1] [2] [1] [1] [1] [1]
[2] [1] [1] [1] [1] [1]
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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/@#&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Array[Length[ptnmats[Times@@Prime/@nrmptn[#]]]&, 30]
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CROSSREFS
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Cf. A007716, A056239, A112798, A120733, A181821, A305936, A318283, A318284, A323295, A323300, A323351.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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