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%I #9 Sep 08 2019 13:07:42
%S 4,16,57,164,403,870,1535,3393,6492,10075,16250,22837,35092,53862,
%T 77929,102925,130837,163010,189773,245903,330117,413691,508391,599788,
%U 680172,767719,864615,945420,1075524,1189739,1217843,1282919,1376563,1465693,1505040
%N Product of the first n 3-almost primes, divided by product of the first n primes, rounded down.
%C Note that this is nonmonotonic. What is the asymptotic value of the ratio A112141(n)/A002110(n)?
%C Probably it can be easily proved that a(n) = 0 for n >= 116. - _Giovanni Resta_, Jun 13 2016
%H Giovanni Resta, <a href="/A122032/b122032.txt">Table of n, a(n) for n = 1..250</a>
%F a(n) = floor(A114425(n)/A002110(n)) = floor(Prod(i=1..n)3almostprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A014612(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A014612(i)/A000040(i))).
%e a(1) = floor(8/2) = floor(4) = 4.
%e a(2) = floor(96/6) = floor(4) = 16.
%e a(3) = floor(1728/30) = floor(57.6) = 57.
%e a(4) = floor(34560/210) = floor(164.571429) = 164.
%e a(5) = floor(933120/2310) = floor(403.948052) = 403.
%e a(6) = floor(26127360/30030) = floor(870.041958) = 870.
%e a(7) = floor(783820800/510510) = floor(1535.36816) = 1535.
%e a(8) = floor(32920473600/9699690) = floor(3393.97172) = 3393.
%e a(9) = floor(1448500838400/223092870) = floor(6492.81547) = 6492.
%e a(10) = floor(65182537728000/6469693230) = floor(10075.058) = 10075.
%e a(11) = floor(3259126886400000/200560490130) = floor(16250.0943) = 16250.
%e a(12) = floor(169474598092800000/7420738134810) = floor(22837.9704) = 22837.
%t tp = Select[Range[1000], PrimeOmega[#] == 3 &]; m = 1; Table[ Floor[m *= tp[[i]] / Prime[i]], {i, Length@ tp}] (* _Giovanni Resta_, Jun 13 2016 *)
%t Floor[#[[1]]/#[[2]]]&/@Module[{nn=200,tap,len},tap=FoldList[ Times, Select[ Range[ nn],PrimeOmega[#]==3&]];len=Length[tap];Thread[{tap, FoldList[Times, Prime[ Range[len]]]}]] (* _Harvey P. Dale_, Sep 08 2019 *)
%Y Cf. A000040, A014612, A002110, A112141, A123389.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Oct 14 2006
%E a(10) corrected by and a(13)-a(35) from _Giovanni Resta_, Jun 13 2016
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