A122032 Product of the first n 3-almost primes, divided by product of the first n primes, rounded down.

4, 16, 57, 164, 403, 870, 1535, 3393, 6492, 10075, 16250, 22837, 35092, 53862, 77929, 102925, 130837, 163010, 189773, 245903, 330117, 413691, 508391, 599788, 680172, 767719, 864615, 945420, 1075524, 1189739, 1217843, 1282919, 1376563, 1465693, 1505040

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)?

Probably it can be easily proved that a(n) = 0 for n >= 116. - Giovanni Resta, Jun 13 2016

Examples

			a(1) = floor(8/2) = floor(4) = 4.
a(2) = floor(96/6) = floor(4) = 16.
a(3) = floor(1728/30) = floor(57.6) = 57.
a(4) = floor(34560/210) = floor(164.571429) = 164.
a(5) = floor(933120/2310) = floor(403.948052) = 403.
a(6) = floor(26127360/30030) = floor(870.041958) = 870.
a(7) = floor(783820800/510510) = floor(1535.36816) = 1535.
a(8) = floor(32920473600/9699690) = floor(3393.97172) = 3393.
a(9) = floor(1448500838400/223092870) = floor(6492.81547) = 6492.
a(10) = floor(65182537728000/6469693230) = floor(10075.058) = 10075.
a(11) = floor(3259126886400000/200560490130) = floor(16250.0943) = 16250.
a(12) = floor(169474598092800000/7420738134810) = floor(22837.9704) = 22837.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A000040, A014612, A002110, A112141, A123389.

Programs

Mathematica

tp = Select[Range[1000], PrimeOmega[#] == 3 &]; m = 1; Table[ Floor[m *= tp[[i]] / Prime[i]], {i, Length@ tp}] (* Giovanni Resta, Jun 13 2016 *)
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 *)

Formula

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))).

Extensions

a(10) corrected by and a(13)-a(35) from Giovanni Resta, Jun 13 2016