A320891 - OEIS

A320891

Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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