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A014612 Numbers that are the product of exactly three (not necessarily distinct) primes. 203
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 proved that for any sufficiently large odd integer n, the equation n = a + b + c has solutions where each of a, b, c are 3-almost primes (A014612). The number of such solutions, where lg x = log (base 2)(x), is (1/2)((((lg n)/log n))^2)/(2 log n))^(1/3))(sigma(n))(n^2)(1+O(1/lg n)) where sigma(n) is a convergent series given by Meng which is > (1/2). - Jonathan Vos Post, Sep 16 2005

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Almost Prime

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.

FORMULA

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

MAPLE

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

MATHEMATICA

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, fQ[ # ] &] (* from 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 *)

PROG

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

(PARI) 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

(Haskell)

a014612 n = a014612_list !! (n-1)

a014612_list = filter ((== 3) . a001222) [1..]

-- Reinhard Zumkeller, Apr 02 2012

CROSSREFS

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. [From Reinhard Zumkeller, Oct 19 2008]

Cf. A007304 (subsequence). [From Alonso del Arte, Aug 09 2011]

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

Sequence in context: A067537 A046339 A145784 * A226527 A212582 A046369

Adjacent sequences:  A014609 A014610 A014611 * A014613 A014614 A014615

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

EXTENSIONS

More terms from Patrick De Geest, Jun 15 1998.

STATUS

approved

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Last modified April 17 20:01 EDT 2014. Contains 240655 sequences.