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A339661
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Number of factorizations of n into distinct squarefree semiprimes.
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13
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1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0
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OFFSET
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1,210
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COMMENTS
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A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
(6*35) (6*10*21) (6*10*77) (6*55*91) (6*10*15*77)
(10*21) (6*14*15) (6*14*55) (6*65*77) (6*10*21*55)
(14*15) (6*22*35) (10*33*91) (6*10*33*35)
(10*14*33) (10*39*77) (6*14*15*55)
(10*21*22) (14*33*65) (6*15*22*35)
(14*15*22) (14*39*55) (10*14*15*33)
(15*22*91) (10*15*21*22)
(15*26*77)
(21*22*65)
(21*26*55)
(22*35*39)
(26*33*35)
(6*35*143)
(10*21*143)
(14*15*143)
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MATHEMATICA
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bfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[bfacs[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[bfacs[n]], {n, 100}]
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PROG
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(PARI)
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CROSSREFS
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A320656 is the not necessarily strict version.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A322794 counts uniform factorizations, such as these.
A339561 lists positions of nonzero terms.
A320655 counts factorizations into semiprimes, with strict case A322353.
The following count vertex-degree partitions and give their Heinz numbers:
The following count partitions of even length and give their Heinz numbers:
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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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