A122609 Product of the first n 5-almost primes (A014614), divided by product of the first n primes, rounded down.

16, 256, 3686, 42130, 413642, 3563691, 25155471, 214483497, 1566662070, 9508018081, 55207846924, 298420794188, 1513939638809, 8555519354201, 45872146324653, 228495219428460, 1045656088909905, 4662597642352366, 19485482684457652, 82333025427285855

Offset: 1

Author

Jonathan Vos Post, Oct 20 2006

Keywords

• easy
• nonn

Comments

This is to 5-almost primes as A122093 is to 4-almost primes as A122032 is to 3-almost primes and as A122019 is to 2-almost primes (semiprimes). Note that these can nonmonotonic (look at the graphs). What is the asymptotic value of the ratio A014614(n)/A002110(n)?

It appears that a(n) = 0 for n >= 11839. - Giovanni Resta, Jun 13 2016

Examples

			a(1) = floor(32/2) = floor 16 = 16.
a(2) = floor(1536/6) = floor(256) = 256.
a(3) = floor(110592/30) = floor(3686.4) = 3686.
a(4) = floor(8847360/210) = floor(42130.2857) = 42130.
a(5) = floor(955514880/2310) = floor(413642.805) = 413642.
a(6) = floor(107017666560/30030) = floor(3563691.86) = 3563691.
a(7) = floor(12842119987200/510510) floor(61152952320/2431) = floor(25155471.95) = 25155471.
a(8) = floor(2080423437926400/9699690) = floor(214483497.712) = 214483497.
a(9) = floor(349511137571635200/223092870) = floor(1566662070.247) = 1566662070.
a(10) = floor(61513960212607795200/6469693230) = floor(9508018081.501) = 9508018081.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..11840

Crossrefs

Cf. A001222, A002110, A008585, A014613, A008587, A086046, A086047, A122093, A112141, A114426, A014614.

Programs

Mathematica

q = Select[Range[900], PrimeOmega[#] == 5 &]; m = 1; Table[ Floor[ m *= q[[i]] / Prime[i]], {i, Length@ q}] (* Giovanni Resta, Jun 13 2016 *)

Formula

a(n) = floor(A122123(n)/A002110(n)) = floor(Prod(i=1..n)5almostprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A014614(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A014614(i)/A000040(i))).

Extensions

a(12) corrected and a(13)-a(20) from Giovanni Resta, Jun 13 2016