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A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

%I A320892 #14 Dec 08 2018 21:03:31
%S A320892 16,64,81,96,144,160,224,256,324,352,384,400,416,486,544,576,608,625,
%T A320892 640,729,736,784,864,896,928,960,992,1024,1184,1215,1296,1312,1344,
%U A320892 1376,1408,1440,1504,1536,1600,1664,1696,1701,1888,1936,1944,1952,2016,2025
%N A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.
%C A320892 A semiprime (A001358) is a product of any two not necessarily distinct primes.
%C A320892 If A025487(k) is in the sequence then so is every number with the same prime signature. - _David A. Corneth_, Oct 23 2018
%C A320892 Numbers for which A001222(n) is even and A322353(n) is zero. - _Antti Karttunen_, Dec 06 2018
%H A320892 Antti Karttunen, <a href="/A320892/b320892.txt">Table of n, a(n) for n = 1..10000</a>
%e A320892 A complete list of all factorizations of 1296 into semiprimes is:
%e A320892   1296 = (4*4*9*9)
%e A320892   1296 = (4*6*6*9)
%e A320892   1296 = (6*6*6*6)
%e A320892 None of these is strict, so 1296 belongs to the sequence.
%t A320892 strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
%t A320892 Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]
%o A320892 (PARI)
%o A320892 A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
%o A320892 isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ _Antti Karttunen_, Dec 06 2018
%Y A320892 Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.
%K A320892 nonn
%O A320892 1,1
%A A320892 _Gus Wiseman_, Oct 23 2018