A182843 - OEIS

%I #22 Aug 03 2014 14:01:28

%S 0,0,1,3,3,6,6,10,10,11,11,16,16,22,22,23,23,30,30,38,38,39,39,48,48,

%T 50,50,51,51,61,61,72,72,73,73,75,75,87,87,88,88,101,101,115,115,116,

%U 116,131,131,134,134,135,135,151,151,153,153,154,154,171,171,189,189,190,190,192,192,211,211

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

%H Fintan Costello, <a href="/A182843/b182843.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n+1) = a(n)+b(n)+c(n), where b(n) is 1 if n is prime, 0 otherwise (sequence A010051)  and c(n) is the number of primes less than the minimum prime factor of n. Since b(2n)=c(2n)=0 for all n>1 we see that a(2n+1)=a(2n) for all n>1. Taking d(n) to represent sequence A038802 we have a(2n)=a(2n-1)+c(2n-1)+d(n-1).

%e Example: For n=4 the only composite integers greater than or equal to 4 all of whose proper divisors are all less than 4 are 4,6, and 9.  Since there are 3 such integers, a(4)=3.

%t Join[{0}, Table[Length[Select[Range[n, n^2], ! PrimeQ[#] && Divisors[#][[-2]] < n &]], {n, 2, 100}]] (* _T. D. Noe_, Feb 28 2011 *)

%Y Cf. A010051, A038802.

%K nonn

%O 1,4

%A _Fintan Costello_, Feb 28 2011