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

%I A122609 #7 Jun 13 2016 08:08:50
%S A122609 16,256,3686,42130,413642,3563691,25155471,214483497,1566662070,
%T A122609 9508018081,55207846924,298420794188,1513939638809,8555519354201,
%U A122609 45872146324653,228495219428460,1045656088909905,4662597642352366,19485482684457652,82333025427285855
%N A122609 Product of the first n 5-almost primes (A014614), divided by product of the first n primes, rounded down.
%C A122609 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)?
%C A122609 It appears that a(n) = 0 for n >= 11839. - _Giovanni Resta_, Jun 13 2016
%H A122609 Giovanni Resta, <a href="/A122609/b122609.txt">Table of n, a(n) for n = 1..11840</a>
%F A122609 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))).
%e A122609 a(1) = floor(32/2) = floor 16 = 16.
%e A122609 a(2) = floor(1536/6) = floor(256) = 256.
%e A122609 a(3) = floor(110592/30) = floor(3686.4) = 3686.
%e A122609 a(4) = floor(8847360/210) = floor(42130.2857) = 42130.
%e A122609 a(5) = floor(955514880/2310) = floor(413642.805) = 413642.
%e A122609 a(6) = floor(107017666560/30030) = floor(3563691.86) = 3563691.
%e A122609 a(7) = floor(12842119987200/510510) floor(61152952320/2431) = floor(25155471.95) = 25155471.
%e A122609 a(8) = floor(2080423437926400/9699690) = floor(214483497.712) = 214483497.
%e A122609 a(9) = floor(349511137571635200/223092870) = floor(1566662070.247) = 1566662070.
%e A122609 a(10) = floor(61513960212607795200/6469693230) = floor(9508018081.501) = 9508018081.
%t A122609 q = Select[Range[900], PrimeOmega[#] == 5 &]; m = 1; Table[ Floor[ m *= q[[i]] / Prime[i]], {i, Length@ q}] (* _Giovanni Resta_, Jun 13 2016 *)
%Y A122609 Cf. A001222, A002110, A008585, A014613, A008587, A086046, A086047, A122093, A112141, A114426, A014614.
%K A122609 easy,nonn
%O A122609 1,1
%A A122609 _Jonathan Vos Post_, Oct 20 2006
%E A122609 a(12) corrected and a(13)-a(20) from _Giovanni Resta_, Jun 13 2016