

A186929


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


1



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
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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(2n1)+b(2n1)(c(2n1)+f(n1)).


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



