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

1,1

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

Links

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

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))).

Example

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.

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 *)

Crossrefs

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

Sequence in context: A197532 A220106 A281064 * A197674 A297220 A297375

Adjacent sequences: A122029 A122030 A122031 * A122033 A122034 A122035

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 14 2006

Extensions

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

Status

approved