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 A014612 Numbers that are the product of exactly three (not necessarily distinct) primes. 284
 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 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 REFERENCES Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211. LINKS 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 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 a(n) = A078840(3,n). - R. J. Mathar, Jan 30 2019 EXAMPLE 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) MAPLE with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # Wesley Ivan Hurt, Feb 05 2014 MATHEMATICA 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, 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 (Scala) 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) (Python) 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 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. - 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. Sequence in context: A046339 A145784 A358763 * 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 September 23 07:35 EDT 2023. Contains 365544 sequences. (Running on oeis4.)