A122019 -id:A122019 - cp's OEIS frontend

A122093 Product of the first n 4-almost primes, divided by product of the first n primes, rounded down.

8, 64, 460, 2633, 12926, 55682, 196527, 837826, 3059886, 9285173, 26956956, 72856639, 184807084, 541527736, 1520886410, 3873955950, 8929796766, 20494615529, 45883467602, 98229395430, 209914872426, 488915652233, 1113313955086, 2451792530303, 5004689907217

Offset: 1

Jonathan Vos Post, Oct 17 2006

This 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 A114426(n)/A002110(n)?

Probably it can be easily proved that a(n) = 0 for n >= 802. - Giovanni Resta, Jun 13 2016

			a(1) = floor(16/2) = floor(8) = 8.
a(2) = floor((16*24)/(2*3)) = floor(384/6) = floor(64) = 64.
a(3) = floor(13824/30) = floor(460.8) = 460.
a(4) = floor(552960/210) = floor(2633.14286) = 2633.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000040, A002110, A014613, A114426, A122019, A122032.

q = Select[Range[1000], PrimeOmega[#] == 4 &]; m = 1; Table[ Floor[ m *= q[[i]]/ Prime[i]], {i, Length@ q}] (* Giovanni Resta, Jun 13 2016 *)

a(n) = floor(A114426(n)/A002110(n)) = floor(Prod(i=1..n)4almostprime(i)/Prod(i=1..n)prime(i)) = floor(Prod(i=1..n)A014613(i)/Prod(i=1..n)A000040(i)) = floor(Prod(i=1..n)(A014613(i)/A000040(i))).

a(11)-a(25) from Giovanni Resta, Jun 13 2016

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

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 20 2006

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

			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.

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

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

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

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))).

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

cp's OEIS Frontend

A122093 Product of the first n 4-almost primes, divided by product of the first n primes, rounded down.

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