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A186929 Number of squarefree composite integers greater than or equal to n whose proper divisors are all less than n.

%I A186929 #13 Aug 03 2014 14:01:28
%S A186929 0,0,0,1,1,3,2,5,5,5,4,8,8,13,12,12,12,18,18,25,25,25,24,32,32,32,31,
%T A186929 31,31,40,39,49,49,49,48,49,49,60,59,59,59,71,70,83,83,83,82,96,96,96,
%U A186929 96,96,96,111,111,112,112,112,111,127,127,144,143,143,143,144,143,161,161,161,160
%N A186929 Number of squarefree composite integers greater than or equal to n whose proper divisors are all less than n.
%H A186929 Fintan Costello, <a href="/A186929/b186929.txt">Table of n, a(n) for n = 1..1000</a>
%F A186929 a(n+1) = a(n)+b(n)(c(n)+d(n)), where b(n) is 1 if n is squarefree, 0 otherwise (sequence A008966), c(n) is 1 if n is composite, 0 otherwise (sequence A066247), and d(n) is the number of primes less than the minimum prime factor of n.   Since d(2n)=0 for all n we see that a(2n+1)=a(2n)+b(2n)c(2n). Taking f(n) to represent sequence A038802 we have a(2n)=a(2n-1)+b(2n-1)(c(2n-1)+f(n-1)).
%e A186929 For n=6 the only squarefree composite integers greater than or equal to 6 all of whose proper divisors are all less than 6 are 6, 10 and 15.  Since there are 3 such integers, a(6)=3.
%t A186929 Join[{0}, Table[Length[Select[Range[n, n^2], SquareFreeQ[#] && ! PrimeQ[#] && Divisors[#][[-2]] < n &]], {n, 2, 100}]] (* _T. D. Noe_, Mar 01 2011 *)
%Y A186929 Cf. A182843.
%K A186929 nonn
%O A186929 1,6
%A A186929 _Fintan Costello_, Mar 01 2011
%E A186929 more