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
Keywords
Examples
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.
Links
- Fintan Costello, Table of n, a(n) for n = 1..1000
Crossrefs
Cf. A182843.
Programs
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Mathematica
Join[{0}, Table[Length[Select[Range[n, n^2], SquareFreeQ[#] && ! PrimeQ[#] && Divisors[#][[-2]] < n &]], {n, 2, 100}]] (* T. D. Noe, Mar 01 2011 *)
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)).
Extensions
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