cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Jonathan Vos Post, Oct 17 2006

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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))).

Extensions

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

A115133 Partial sums of A064061.

Original entry on oeis.org

429, 1859, 5291, 12363, 25623, 48879, 87639, 149655, 245586, 389796, 601304, 904904, 1332474, 1924494, 2731794, 3817554, 5259579, 7152873, 9612537, 12777017, 16811729, 21913089, 28312977, 36283665, 46143240, 58261554
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2006

Keywords

Links

Crossrefs

Cf. A114426 (sixth column of A115127).

Programs

  • Mathematica
    Accumulate[Table[Binomial[n,7]-Binomial[n,5],{n,13,50}]] (* or *) LinearRecurrence[ {9,-36,84,-126,126,-84,36,-9,1},{429,1859,5291,12363,25623,48879,87639,149655,245586},40] (* Harvey P. Dale, Sep 03 2015 *)

Formula

G.f.:(429-2002*x+4004*x^2-4368*x^3+2730*x^4-924*x^5+132*x^6)/(1-x)^9.
a(n)=A115127(n+7, 7), n>=1, a(0):=C(7)=429, with C(n):=A000108(n) (Catalan).

A122123 Product of the first n 5-almost primes (A014614).

Original entry on oeis.org

32, 1536, 110592, 8847360, 955514880, 107017666560, 12842119987200, 2080423437926400, 349511137571635200, 61513960212607795200, 11072512838269403136000, 2214502567653880627200000, 460616534072007170457600000, 111929817779497742421196800000
Offset: 1

Views

Author

Jonathan Vos Post, Oct 19 2006

Keywords

Comments

5-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086047 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008587(n) = 5*n.

Examples

			a(10) = 32 * 48 * 72 * 80 * 108 * 112 * 120 * 162 * 168 * 176 = 2^33 * 3^12 * 5^2 * 7^2 * 11 which has 50 prime factors with multiplicity.

Crossrefs

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

Programs

  • Mathematica
    Rest[FoldList[Times,1,Select[Range[200],PrimeOmega[#]==5&]]] (* Harvey P. Dale, Feb 07 2012 *)

Formula

a(n) = Prod[i=1..n] A014614(i).

Extensions

More terms from Harvey P. Dale, Feb 07 2012

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

Views

Author

Jonathan Vos Post, Oct 20 2006

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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))).

Extensions

a(12) corrected and a(13)-a(20) from Giovanni Resta, Jun 13 2016
Showing 1-4 of 4 results.

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