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A320891 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

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.

MATHEMATICA

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[#]=={}]&]

CROSSREFS

Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320892, A320893, A320894.

Sequence in context: A320894 A122986 A066427 * A161697 A078593 A168350

Adjacent sequences:  A320888 A320889 A320890 * A320892 A320893 A320894

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 23 2018

STATUS

approved

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Last modified March 26 14:18 EDT 2019. Contains 321497 sequences. (Running on oeis4.)