A320891 - OEIS

%I #13 Feb 07 2021 07:05:28

%S 4,9,16,24,25,40,49,54,56,64,81,88,96,104,121,135,136,144,152,160,169,

%T 184,189,224,232,240,248,250,256,289,296,297,324,328,336,344,351,352,

%U 361,375,376,384,400,416,424,459,472,486,488,513,528,529,536,544,560

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

%C A squarefree semiprime (A006881) is a product of any two distinct primes.

%C Also numbers with an even number x of prime factors, whose greatest prime multiplicity exceeds x/2.

%H Amiram Eldar, <a href="/A320891/b320891.txt">Table of n, a(n) for n = 1..10000</a>

%e A complete list of all factorizations of 24 is:

%e   (2*2*2*3),

%e   (2*2*6), (2*3*4),

%e   (2*12), (3*8), (4*6),

%e   (24).

%e All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.

%t 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 Select[Range[100],And[EvenQ[PrimeOmega[#]],semfacs[#]=={}]&]

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 23 2018