login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A320891
Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.
28
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
1,1
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
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[#]=={}]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 23 2018
STATUS
approved