A086046 -id:A086046 - cp's OEIS frontend

A114426 Product of the first n 4-almost primes (A014613).

16, 384, 13824, 552960, 29859840, 1672151040, 100329062400, 8126654054400, 682638940569600, 60072226770124800, 5406500409311232000, 540650040931123200000, 56227604256836812800000

Offset: 1

Jonathan Vos Post, Feb 13 2006

4-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086046 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008586(n) = 4*n.

			a(5) = 29859840 = 16 * 24 * 36 * 40 * 54 = the product of the first 5 values of the 4-almost primes = 2^13 * 3^6 * 5, which has 4*5 = 20 prime factors (with multiplicity).

Cf. A001222, A002110, A008585, A014613, A008586, A086046, A112141.

FoldList[Times,Select[Range[200],PrimeOmega[#]==4&]] (* Harvey P. Dale, Dec 02 2018 *)

a(n) = Prod[from i = 1 to n] A014613(i).

A110226 1 + sum of first n 4-almost primes.

Original entry on oeis.org

1, 17, 41, 77, 117, 171, 227, 287, 368, 452, 540, 630, 730, 834, 960, 1092, 1227, 1363, 1503, 1653, 1805, 1961, 2145, 2334, 2530, 2728, 2932, 3142, 3362, 3587, 3815, 4047, 4281, 4529, 4779, 5039, 5315, 5609, 5905, 6202, 6508, 6816, 7131, 7459, 7789, 8129

Offset: 0

Jonathan Vos Post, Sep 06 2005

First differences are the sequence of 4-almost primes (A014613). Hence a(n) is the least positive sequence whose first differences are the sequence of 4-almost primes. Primes in this sequence include: a(1) = 17, a(2) = 41, a(6) = 227, a(35) = 5039, a(43) = 7459, a(44) = 7789. Semiprimes in this sequence include: a(3) = 77 = 7 * 11, a(7) = 287 = 7 * 41, a(16) = 1227 = 3 * 409, a(17) = 1363 = 29 * 47, a(21) = 1961 = 37 * 53, a(27) = 3142 = 2 * 1571, a(29) = 3587 = 17 * 211, a(32) = 4281 = 3 * 1427, a(33) = 4529 = 7 * 647, a(36) = 5315 = 5 * 1063, a(37) = 5609 = 71 * 79, a(38) = 5905 = 5 * 1181, a(42) = 7131 = 3 * 2733a(45) = 8129 = 11 * 739. 3-almost primes in this sequence include: a(4) = 117 = 3^2 * 13, a(5) = 171 = 3^2 * 19, a(9) = 452 = 2^2 * 113, a(12) = 730 = 2 * 5 * 73, a(13) = 833 = 2 * 3 * 139, a(18) = 3^2 * 167, a(19) = 1653 = 3 * 19 * 29, a(20) = 1805 = 5 * 19^2, a(23) = 2534 = 2 * 3 * 389, a(26) = 2932 = 2^2 * 733, a(28) = 3362 = 2 * 41^2, a(30) = 3815 = 5 * 7 * 109, a(31) = 4047 = 3 * 19 * 71, a(39) = 2 * 7 * 443, a(40) = 6508 = 2^2 * 1627. 4-almost primes in this sequence include: a(22) = 2145 = 3 * 5 * 11 * 13, a(24) = 2530 = 2 * 5 * 11 * 23.

Cf. A014613, A086046, A110208, A110209.

Accumulate[Join[{1},Select[Range[500],PrimeOmega[#]==4&]]] (* Harvey P. Dale, Dec 13 2018 *)

a(0) = 1; for n>0, a(n) = 1 + A086046(n).

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

Jonathan Vos Post, Oct 19 2006

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.

			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.

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

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

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

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

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

A382831 a(n) is the n-th n-almost-prime that is a partial sum of the sequence of n-almost-primes.

Original entry on oeis.org

2, 10, 964, 1804, 7820, 48120, 830817, 4895208, 11308160, 162802560, 394129476, 3763612800, 19823090472, 1018716103620, 9744542956800, 3989325082624, 329306801920000, 2978224618328064, 11804664377696256, 128906665137012736

Offset: 1

Robert Israel, Apr 28 2025

nonn

			The first three members of A086062 that are 3-almost-primes are 8 = 2^3, 20 = 2^2 * 5 = 8 + 12, and 964 = 2^2 * 241 = 8 + 12 + 18 + ... + 92, so a(3) = 964.

Cf. A007504, A062198, A086062, A086046, A086047, A086052, A086059, A086061.

f:= proc(n) uses priqueue;
  local pq,t,s,x,p,i,count;
  initialize(pq);
  insert([-2^n,2$n],pq);
  s:= 0; count:= 0:
  do
    t:= extract(pq);
    x:= -t[1];
    s:= s + x;
    if numtheory:-bigomega(s) = n  then count:= count+1; if count = n then return s fi fi;
    p:= nextprime(t[-1]);
    for i from n+1 to 2 by -1 while t[i] = t[-1] do
          insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
    od;
  od
end proc:
map(f, [$1..20]);

cp's OEIS Frontend

A114426 Product of the first n 4-almost primes (A014613).

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A110226 1 + sum of first n 4-almost primes.

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

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

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A382831 a(n) is the n-th n-almost-prime that is a partial sum of the sequence of n-almost-primes.

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