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A320891
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Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.
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28
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4, 9, 16, 24, 25, 40, 49, 54, 56, 64, 81, 88, 96, 104, 121, 135, 136, 144, 152, 160, 169, 184, 189, 224, 232, 240, 248, 250, 256, 289, 296, 297, 324, 328, 336, 344, 351, 352, 361, 375, 376, 384, 400, 416, 424, 459, 472, 486, 488, 513, 528, 529, 536, 544, 560
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OFFSET
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1,1
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COMMENTS
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A squarefree semiprime (A006881) is a product of any two distinct primes.
Also numbers with an even number x of prime factors, whose greatest prime multiplicity exceeds x/2.
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LINKS
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EXAMPLE
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A complete list of all factorizations of 24 is:
(2*2*2*3),
(2*2*6), (2*3*4),
(2*12), (3*8), (4*6),
(24).
All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.
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MATHEMATICA
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semfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[semfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], And[SquareFreeQ[#], PrimeOmega[#]==2]&]}]];
Select[Range[100], And[EvenQ[PrimeOmega[#]], semfacs[#]=={}]&]
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CROSSREFS
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Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320892, A320893, A320894.
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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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