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A320891 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.

%I A320891 #13 Feb 07 2021 07:05:28
%S A320891 4,9,16,24,25,40,49,54,56,64,81,88,96,104,121,135,136,144,152,160,169,
%T A320891 184,189,224,232,240,248,250,256,289,296,297,324,328,336,344,351,352,
%U A320891 361,375,376,384,400,416,424,459,472,486,488,513,528,529,536,544,560
%N A320891 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.
%C A320891 A squarefree semiprime (A006881) is a product of any two distinct primes.
%C A320891 Also numbers with an even number x of prime factors, whose greatest prime multiplicity exceeds x/2.
%H A320891 Amiram Eldar, <a href="/A320891/b320891.txt">Table of n, a(n) for n = 1..10000</a>
%e A320891 A complete list of all factorizations of 24 is:
%e A320891   (2*2*2*3),
%e A320891   (2*2*6), (2*3*4),
%e A320891   (2*12), (3*8), (4*6),
%e A320891   (24).
%e A320891 All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.
%t A320891 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]&]}]];
%t A320891 Select[Range[100],And[EvenQ[PrimeOmega[#]],semfacs[#]=={}]&]
%Y A320891 Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320892, A320893, A320894.
%K A320891 nonn
%O A320891 1,1
%A A320891 _Gus Wiseman_, Oct 23 2018