A122019 Product of the first n semiprimes, divided by product of the first n primes, rounded down.

2, 4, 7, 10, 13, 15, 18, 21, 23, 21, 22, 20, 17, 15, 12, 11, 9, 7, 6, 5, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Jonathan Vos Post, Oct 14 2006

Keywords

• easy
• nonn

Comments

Note that this is nonmonotonic. What is the asymptotic value of the ratio A112141(n)/A002110(n)?

Examples

			a(1) = floor(4/2) = floor(2) = 2.
a(2) = floor(24/6) = floor(4) = 4.
a(3) = floor(216/30) = floor(7.2) = 7.
a(4) = floor(2160/210) = floor(10.2857143) = 10.
a(5) = floor(30240/2310) = floor(13.0909090909) = 13.
a(6) = floor(453600/30030) = floor(15.1048951) = 15.
a(7) = floor(9525600/510510) = floor(18.6589881) = 18.
a(8) = floor(209563200/9699690) = floor(21.6051441) = 21.
a(9) = floor(5239080000/223092870) = floor(23.4838523) = 23.
a(10) = floor(136216080000/6469693230) = floor(21.0544882) = 21.
a(11) = floor(4495130640000/200560490130) = floor(22.4128423) = 22.
a(12) = floor(152834441760000/7420738134810) = floor(20.5955848) = 20.
		

Crossrefs

Cf. A000040, A001358, A002110, A112141, A123389.

Programs

Mathematica

sp = Select[Range@ 250, PrimeOmega[#] == 2 &]; m = 1; Table[ Floor[m *= sp[[i]] / Prime[i]], {i, Length@ sp}] (* Giovanni Resta, Jun 13 2016 *)

Formula

a(n) = floor(A112141(n)/A002110(n)) = floor(Prod(i=1..n)semiprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A001358(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A001358(i)/A000040(i))).

Extensions

a(13)-a(82) from Giovanni Resta, Jun 13 2016