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

Offset: 1

Author

Gus Wiseman, Oct 23 2018

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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[#]=={}]&]