A086062 -id:A086062 - cp's OEIS frontend

A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244

Offset: 1

Eric W. Weisstein

nonn

Sometimes called "triprimes" or "3-almost primes".
See also A001358 for product of two primes (sometimes called semiprimes).
If you graph a(n)/n for n up to 10000 (and probably quite a bit higher), it appears to be converging to something near 3.9. In fact the limit is infinite. - Franklin T. Adams-Watters, Sep 20 2006
Meng shows that for any sufficiently large odd integer n, the equation n = a + b + c has solutions where each of a, b, c is 3-almost prime. The number of such solutions is (log log n)^6/(16 (log n)^3)*n^2*s(n)*(1 + O(1/log log n)), where s(n) = Sum_{q >= 1} Sum_{a = 1..q, (a, q) = 1} exp(i*2*Pi*n*a/q)*mu(n)/phi(n)^3 > 1/2. - Jonathan Vos Post, Sep 16 2005, corrected & rewritten by M. F. Hasler, Apr 24 2019
Also, a(n) are the numbers such that exactly half of their divisors are composite. For the numbers in which exactly half of the divisors are prime, see A167171. - Ivan Neretin, Jan 12 2016

			From _Gus Wiseman_, Nov 04 2020: (Start)
Also Heinz numbers of integer partitions into three parts, counted by A001399(n-3) = A069905(n) with ordered version A000217, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence of terms together with their prime indices begins:
      8: {1,1,1}     70: {1,3,4}     130: {1,3,6}
     12: {1,1,2}     75: {2,3,3}     138: {1,2,9}
     18: {1,2,2}     76: {1,1,8}     147: {2,4,4}
     20: {1,1,3}     78: {1,2,6}     148: {1,1,12}
     27: {2,2,2}     92: {1,1,9}     153: {2,2,7}
     28: {1,1,4}     98: {1,4,4}     154: {1,4,5}
     30: {1,2,3}     99: {2,2,5}     164: {1,1,13}
     42: {1,2,4}    102: {1,2,7}     165: {2,3,5}
     44: {1,1,5}    105: {2,3,4}     170: {1,3,7}
     45: {2,2,3}    110: {1,3,5}     171: {2,2,8}
     50: {1,3,3}    114: {1,2,8}     172: {1,1,14}
     52: {1,1,6}    116: {1,1,10}    174: {1,2,10}
     63: {2,2,4}    117: {2,2,6}     175: {3,3,4}
     66: {1,2,5}    124: {1,1,11}    182: {1,4,6}
     68: {1,1,7}    125: {3,3,3}     186: {1,2,11}
(End)

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211.

T. D. Noe, Table of n, a(n) for n = 1..10000
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
Eric Weisstein's World of Mathematics, Almost Prime

Cf. A000040, A001358 (biprimes), A014613 (quadruprimes), A033942, A086062, A098238, A123072, A123073, A101605 (characteristic function).
Cf. A109251 (number of 3-almost primes <= 10^n).
Subsequence of A145784. - Reinhard Zumkeller, Oct 19 2008
Cf. A007304 is the squarefree case.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), this sequence (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Cf. A253721 (final digits).
A014311 is a different ranking of ordered triples, with strict case A337453.
A046316 is the restriction to odds, with strict case A307534.
A075818 is the restriction to evens, with strict case A075819.
A285508 is the nonsquarefree case.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
Cf. A000212, A000217, A046389, A140106, A307719, A321773.

a014612 n = a014612_list !! (n-1)
a014612_list = filter ((== 3) . a001222) [1..] -- Reinhard Zumkeller, Apr 02 2012

with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # Wesley Ivan Hurt, Feb 05 2014

threeAlmostPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, threeAlmostPrimeQ[ # ] &] (* Robert G. Wilson v, Jan 04 2006 *)
NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; NestList[NextkAlmostPrime[#, 3] &, 2^3, 56] (* Robert G. Wilson v, Jan 27 2013 *)
Select[Range[244], PrimeOmega[#] == 3 &] (* Jayanta Basu, Jul 01 2013 *)

isA014612(n)=bigomega(n)==3 \\ Charles R Greathouse IV, May 07 2011

list(lim)=my(v=List(),t);forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q),listput(v,t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 04 2013

from sympy import factorint
def ok(n): f = factorint(n); return sum(f[p] for p in f) == 3
print(list(filter(ok, range(245)))) # Michael S. Branicky, Aug 12 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014612(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

def primeFactors(number: Int, list: List[Int] = List())
                                                      : List[Int] = {
  for (n <- 2 to number if (number % n == 0)) {
    return primeFactors(number / n, list :+ n)
  }
  list
}
(1 to 250).filter(primeFactors().size == 3) // _Alonso del Arte, Nov 04 2020, based on algorithm by Victor Farcic (vfarcic)

Product p_i^e_i with Sum e_i = 3.
a(n) ~ 2n log n / (log log n)^2 as n -> infinity [Landau, p. 211].
Tau(a(n)) = 2 * (omega(a(n)) + 1) = 2*A083399(a(n)), where tau = A000005 and omega = A001221. - Wesley Ivan Hurt, Jun 28 2013
a(n) = A078840(3,n). - R. J. Mathar, Jan 30 2019

More terms from Patrick De Geest, Jun 15 1998

A110209 1 + sum of first n 3-almost primes.

Original entry on oeis.org

1, 9, 21, 39, 59, 86, 114, 144, 186, 230, 275, 325, 377, 440, 506, 574, 644, 719, 795, 873, 965, 1063, 1162, 1264, 1369, 1479, 1593, 1709, 1826, 1950, 2075, 2205, 2343, 2490, 2638, 2791, 2945, 3109, 3274, 3444, 3615, 3787, 3961, 4136, 4318, 4504, 4692

Offset: 0

Jonathan Vos Post, Sep 06 2005

First differences are the sequence of 3-almost primes (A014612). Hence a(n) is the least positive sequence whose first differences are the sequence of 3-almost primes. Primes in this sequence include: a(1) = a(4) = 59, a(17) = 719, a(21) = 1063, a(27) = 1709, a(35) = 2791, a(37) = 3109, . Semiprimes in this sequence include: a(1) = 3^2, a(2) = 21 = 3 * 7, a(3) = 39 = 3 * 13, a(5) = 86 = 2 * 43, a(12) = 377 = 13 * 29, a(20) = 965 = 5 * 193, a(24) = 1369 = 37^2, a(34) = 2638 = 2 * 1319, a(38) = 3274 = 2 * 1637, a(41) = 3787 = 7 * 541, a(42) = 3961 = 17 * 223, a(47) = 4882 = 2 * 2441. 3-almost primes in this sequence include: a(1) = 2^3, a(6) = 114 = 2 * 3 * 19, a(8) = 186 = 2 * 3 * 31, a(9) = 230 = 2 * 5 * 23, a(10) = 275 = 5^2 * 11, a(11) = 325 = 5^2 * 13, a(14) = 506 = 2 * 11 * 23, a(15) = 574 = 2 * 7 * 41, a(18) = 795 = 3 * 5 * 53, a(19) = 873 + 3^2 * 97, a(22) = 1162 = 2 * 7 * 83, a(25) = 1479 = 3 * 17 * 29, a(28) = 1826 = 2 * 11 * 83, a(30) = 2075 = 5^2 * 83, a(32) = 2343 = 3 * 11 * 71, a(36) = 2945 = 5 * 19 * 31, a(40) = 3615 = 3 * 5 * 241, a(44) = 4318 = 2 * 17 * 127. Note also the powers a(7) = 144 = 12^2.

Cf. A014612, A086062, A110208, A110226.

a(0) = 1; for n>0, a(n) = 1 + A086062(n).

A114425 Product of the first n 3-almost primes (A014612).

Original entry on oeis.org

8, 96, 1728, 34560, 933120, 26127360, 783820800, 32920473600, 1448500838400, 65182537728000, 3259126886400000, 169474598092800000, 10676899679846400000, 704675378869862400000, 47917925763150643200000

Offset: 1

Jonathan Vos Post, Feb 13 2006

3-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086062 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008585(n) = 3*n.

			a(5) = 933120 = 8 * 12 * 18 * 20 * 27 = the product of the first 5 values of the 3-almost primes = 2^8 * 3^6 * 5, which has 3*5 = 15 prime factors (with multiplicity).
a(20) = 137199755075271237225676800000000 = 8 * 12 * 18 * 20 * 27 * 28 * 30 * 42 * 44 * 45 * 50 * 52 * 63 * 66 * 68 * 70 * 75 * 76 * 78 * 92 = 2^26 * 3^15 * 5^8 * 7^4 * 11, which has 20*3 = 60 prime factors (with multiplicity).

Harvey P. Dale, Table of n, a(n) for n = 1..364

Cf. A002110, A008585, A014612, A086062, A112141.

FoldList[Times,Select[Range[70],PrimeOmega[#]==3&]] (* Harvey P. Dale, Apr 26 2020 *)

a(n) = Prod[from i = 1 to n] A014612(i).

A217018 Partial sums of 3-almost primes which are again 3-almost primes, i.e., have exactly 3 not necessarily distinct prime factors.

Original entry on oeis.org

8, 20, 964, 1825, 2074, 2637, 3614, 3786, 4503, 5283, 5495, 6414, 6652, 7138, 7383, 9485, 9764, 10330, 10615, 11191, 12427, 12749, 13074, 15475, 16195, 16930, 18446, 19233, 20855, 22108, 22959, 23387, 28273, 28747, 29222, 30676, 32695, 34798, 35871

Offset: 1

M. F. Hasler, Sep 23 2012

nonn

Bigomega=3 analog of the semiprime version A092190. In sequence A086062 it was asked whether there are infinitely many such numbers.

A217018(n,list=0,N=0,S=0)={until(!n--,until(bigomega(S+=N)==3,until(bigomega(N++)==3,));list&print1(S","));S} \\ - M. F. Hasler, Sep 29 2012

A217018 = A086062 intersect A014612.

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]);

A216686 Numbers n such that n appears in the partial sums of the m-almost primes, where m=bigomega(n).

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 20, 32, 40, 41, 58, 64, 80, 128, 160, 185, 197, 219, 254, 256, 281, 320, 377, 512, 589, 640, 843, 917, 964, 1024, 1247, 1280, 1652, 1707, 1804, 1825, 2048, 2074, 2157, 2519, 2560, 2637, 2642, 2727, 2771, 3614, 3755, 3786, 4046, 4096, 4227

Offset: 1

Gerasimov Sergey, Sep 13 2012

A013918 is a subsequence. - Zak Seidov, Sep 17 2012

Or: Numbers n equal to the sum of the first k numbers x having bigomega(x)=bigomega(n), for some k. - M. F. Hasler, Sep 23 2012

			2 is in the sequence because 2 appears in A007504.
4 is in the sequence because 4 appears in A062198.
5 is in the sequence because 5 appears in A007504.
6 is not in the sequence because 6 is not in A062198.
8 is in the sequence because 8 appears in A086062,
10 is in the sequence because 10 appears in A062198.

Cf. A001222, A007504, A013918, A062198, A092190, A086052, A086062.

alm := proc(n,m) # n-th m-almost prime
    option remember;
    if n =1 then
        2^m ;
    else
        for a from procname(n-1,m)+1 do
            if numtheory[bigomega](a) = m then
                return a;
            end if;
        end do:
    end if;
end proc:
almP := proc(n,m) #n-th partial sum of the m-almost primes
    add(alm(i,m),i=1..n) ;
end proc:
isA216686 := proc(n) # is n in the sequence?
    local m ,k,ps;
    m := numtheory[bigomega](n) ;
    for k from 1 do
        ps := almP(k,m) ;
        if ps = n then
            return true;
        elif ps > n then
            return false;
        end  if;
    end do:
end proc:
for n from 1 to 4300 do
    if isA216686(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 14 2012

is_A216686(n)={ my(m=bigomega(n),t); while(n>0, while(bigomega(t++)!=m,); n-=t); !n}  \\ - M. F. Hasler, Sep 23 2012

Corrected by R. J. Mathar, Sep 14 2012

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A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

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A110209 1 + sum of first n 3-almost primes.

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A114425 Product of the first n 3-almost primes (A014612).

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A217018 Partial sums of 3-almost primes which are again 3-almost primes, i.e., have exactly 3 not necessarily distinct prime factors.

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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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A216686 Numbers n such that n appears in the partial sums of the m-almost primes, where m=bigomega(n).

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