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

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, 31, 31, 40, 39, 49, 49, 49, 48, 49, 49, 60, 59, 59, 59, 71, 70, 83, 83, 83, 82, 96, 96, 96, 96, 96, 96, 111, 111, 112, 112, 112, 111, 127, 127, 144, 143, 143, 143, 144, 143, 161, 161, 161, 160

Offset

1,6

Links

Fintan Costello, Table of n, a(n) for n = 1..1000

Formula

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)).

Example

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.

Mathematica

Join[{0}, Table[Length[Select[Range[n, n^2], SquareFreeQ[#] && ! PrimeQ[#] && Divisors[#][[-2]] < n &]], {n, 2, 100}]] (* T. D. Noe, Mar 01 2011 *)

Crossrefs

Cf. A182843.

Sequence in context: A001598 A141297 A303917 * A223538 A059319 A196438

Adjacent sequences: A186926 A186927 A186928 * A186930 A186931 A186932

Keyword

nonn

Author

Fintan Costello, Mar 01 2011

Extensions

more

Status

approved