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A014612
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Numbers that are the product of exactly three (not necessarily distinct) primes.
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194
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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
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OFFSET
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1,1
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COMMENTS
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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
Triprime characteristic: floor(Omega(n)/3) * floor(3/Omega(n)). - Wesley Ivan Hurt, Jan 10 2013
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REFERENCES
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E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
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LINKS
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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.
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FORMULA
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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].
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000040, A001358 (biprimes), A014613 (quadruprimes), A033942, A086062, A098238, A123072, A123073.
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 * A212582 A046369 A066428
Adjacent sequences: A014609 A014610 A014611 * A014613 A014614 A014615
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KEYWORD
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nonn,changed
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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More terms from Patrick De Geest, Jun 15 1998.
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STATUS
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approved
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