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A340853
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Number of factorizations of n such that every factor is a multiple of the number of factors.
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8
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0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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Also factorizations whose greatest common divisor is a multiple of the number of factors.
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LINKS
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EXAMPLE
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The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
2 4 16 48 96 144 216 240 432
2*2 2*8 6*8 2*48 2*72 4*54 4*60 6*72
4*4 2*24 4*24 4*36 6*36 6*40 8*54
4*12 6*16 6*24 12*18 8*30 12*36
8*12 8*18 2*108 10*24 18*24
12*12 6*6*6 12*20 2*216
3*3*24 2*120 4*108
3*6*12 3*3*48
3*6*24
6*6*12
3*12*12
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], n>1&&Divisible[GCD@@#, Length[#]]&]], {n, 100}]
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CROSSREFS
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Positions of terms > 1 are A100716.
The version for strict partitions is A340830.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
- Factorizations -
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even factors, even-length case A340786.
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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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