cp's OEIS Frontend

A224705 Composite numbers n divisible by Omega(n)^2 (the square of the number of their prime factors, counted with multiplicity).

%I A224705 #18 Mar 14 2016 12:13:11
%S A224705 4,16,18,27,45,63,99,117,144,153,171,200,207,216,256,261,279,300,324,
%T A224705 333,360,369,384,387,423,450,477,500,504,531,540,549,576,603,639,640,
%U A224705 657,675,700,711,747,750,756,792,801,873,896,900,909,927,936,960,963,981
%N A224705 Composite numbers n divisible by Omega(n)^2 (the square of the number of their prime factors, counted with multiplicity).
%C A224705 A number n is in the sequence if and only if mod(n, A001222(n)^2) == 0 and n is not prime.
%C A224705 Without the restriction that n must be composite, all prime numbers would trivially be included in the sequence.
%H A224705 Christian N. K. Anderson, <a href="/A224705/b224705.txt">Table of n, a(n) for n = 1..10000</a>
%e A224705 a(6)=63=3*3*7, and 63 is divisible by 9=3^2; a(9)=144, which has 6 prime factors and is divisible by 36.
%p A224705 isA224705 := proc(n)
%p A224705     if isprime(n) then
%p A224705         return false;
%p A224705     else
%p A224705         if modp(n,numtheory[bigomega](n)^2) = 0 then
%p A224705             true;
%p A224705         else
%p A224705             false;
%p A224705         end if;
%p A224705     end if;
%p A224705 end proc:
%p A224705 n := 1;
%p A224705 c := 4;
%p A224705 while n <= 10000 do
%p A224705     if isA224705(c) then
%p A224705         printf("%d %d\n",n,c) ;
%p A224705         n := n+1 ;
%p A224705     end if;
%p A224705     c := c+1 ;
%p A224705 end do: # _R. J. Mathar_, Mar 14 2016
%t A224705 Select[Range[2, 1000], ! PrimeQ[#] && Mod[#, PrimeOmega[#]^2] == 0 &] (* _T. D. Noe_, Apr 18 2013 *)
%o A224705 (R) y=c(); i=2; isint<-function(x) x==as.integer(x)
%o A224705 while(length(y)<10000) {Omega=length(factorize(i)); if(Omega>1) if(isint(i/Omega^2)) y=c(y,i); i=i+1 }
%Y A224705 Cf. A001222, A074946, A137230, A070003, A224703.
%K A224705 nonn
%O A224705 1,1
%A A224705 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Apr 16 2013