This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A014612 #148 Feb 16 2025 08:32:33
%S A014612 8,12,18,20,27,28,30,42,44,45,50,52,63,66,68,70,75,76,78,92,98,99,102,
%T A014612 105,110,114,116,117,124,125,130,138,147,148,153,154,164,165,170,171,
%U A014612 172,174,175,182,186,188,190,195,207,212,222,230,231,236,238,242,244
%N A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.
%C A014612 Sometimes called "triprimes" or "3-almost primes".
%C A014612 See also A001358 for product of two primes (sometimes called semiprimes).
%C A014612 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
%C A014612 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
%C A014612 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
%D A014612 Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211.
%H A014612 T. D. Noe, <a href="/A014612/b014612.txt">Table of n, a(n) for n = 1..10000</a>
%H A014612 Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http://name.umdl.umich.edu/ABV2766.0001.001">vol. 1</a> and <a href="http://name.umdl.umich.edu/ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
%H A014612 Xianmeng Meng, <a href="http://dx.doi.org/10.1016/j.jnt.2005.04.013">On sums of three integers with a fixed number of prime factors</a>, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
%H A014612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>
%F A014612 Product p_i^e_i with Sum e_i = 3.
%F A014612 a(n) ~ 2n log n / (log log n)^2 as n -> infinity [Landau, p. 211].
%F A014612 Tau(a(n)) = 2 * (omega(a(n)) + 1) = 2*A083399(a(n)), where tau = A000005 and omega = A001221. - _Wesley Ivan Hurt_, Jun 28 2013
%F A014612 a(n) = A078840(3,n). - _R. J. Mathar_, Jan 30 2019
%e A014612 From _Gus Wiseman_, Nov 04 2020: (Start)
%e A014612 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:
%e A014612 8: {1,1,1} 70: {1,3,4} 130: {1,3,6}
%e A014612 12: {1,1,2} 75: {2,3,3} 138: {1,2,9}
%e A014612 18: {1,2,2} 76: {1,1,8} 147: {2,4,4}
%e A014612 20: {1,1,3} 78: {1,2,6} 148: {1,1,12}
%e A014612 27: {2,2,2} 92: {1,1,9} 153: {2,2,7}
%e A014612 28: {1,1,4} 98: {1,4,4} 154: {1,4,5}
%e A014612 30: {1,2,3} 99: {2,2,5} 164: {1,1,13}
%e A014612 42: {1,2,4} 102: {1,2,7} 165: {2,3,5}
%e A014612 44: {1,1,5} 105: {2,3,4} 170: {1,3,7}
%e A014612 45: {2,2,3} 110: {1,3,5} 171: {2,2,8}
%e A014612 50: {1,3,3} 114: {1,2,8} 172: {1,1,14}
%e A014612 52: {1,1,6} 116: {1,1,10} 174: {1,2,10}
%e A014612 63: {2,2,4} 117: {2,2,6} 175: {3,3,4}
%e A014612 66: {1,2,5} 124: {1,1,11} 182: {1,4,6}
%e A014612 68: {1,1,7} 125: {3,3,3} 186: {1,2,11}
%e A014612 (End)
%p A014612 with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # _Wesley Ivan Hurt_, Feb 05 2014
%t A014612 threeAlmostPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, threeAlmostPrimeQ[ # ] &] (* _Robert G. Wilson v_, Jan 04 2006 *)
%t A014612 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 *)
%t A014612 Select[Range[244], PrimeOmega[#] == 3 &] (* _Jayanta Basu_, Jul 01 2013 *)
%o A014612 (PARI) isA014612(n)=bigomega(n)==3 \\ _Charles R Greathouse IV_, May 07 2011
%o A014612 (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
%o A014612 (Haskell) a014612 n = a014612_list !! (n-1)
%o A014612 a014612_list = filter ((== 3) . a001222) [1..] -- _Reinhard Zumkeller_, Apr 02 2012
%o A014612 (Scala) def primeFactors(number: Int, list: List[Int] = List())
%o A014612 : List[Int] = {
%o A014612 for (n <- 2 to number if (number % n == 0)) {
%o A014612 return primeFactors(number / n, list :+ n)
%o A014612 }
%o A014612 list
%o A014612 }
%o A014612 (1 to 250).filter(primeFactors(_).size == 3) // _Alonso del Arte_, Nov 04 2020, based on algorithm by Victor Farcic (vfarcic)
%o A014612 (Python)
%o A014612 from sympy import factorint
%o A014612 def ok(n): f = factorint(n); return sum(f[p] for p in f) == 3
%o A014612 print(list(filter(ok, range(245)))) # _Michael S. Branicky_, Aug 12 2021
%o A014612 (Python)
%o A014612 from math import isqrt
%o A014612 from sympy import primepi, primerange, integer_nthroot
%o A014612 def A014612(n):
%o A014612 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)))
%o A014612 m, k = n, f(n)
%o A014612 while m != k:
%o A014612 m, k = k, f(k)
%o A014612 return m # _Chai Wah Wu_, Aug 17 2024
%Y A014612 Cf. A000040, A001358 (biprimes), A014613 (quadruprimes), A033942, A086062, A098238, A123072, A123073, A101605 (characteristic function).
%Y A014612 Cf. A109251 (number of 3-almost primes <= 10^n).
%Y A014612 Subsequence of A145784. - _Reinhard Zumkeller_, Oct 19 2008
%Y A014612 Cf. A007304 is the squarefree case.
%Y A014612 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
%Y A014612 Cf. A253721 (final digits).
%Y A014612 A014311 is a different ranking of ordered triples, with strict case A337453.
%Y A014612 A046316 is the restriction to odds, with strict case A307534.
%Y A014612 A075818 is the restriction to evens, with strict case A075819.
%Y A014612 A285508 is the nonsquarefree case.
%Y A014612 A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
%Y A014612 Cf. A000212, A000217, A046389, A140106, A307719, A321773.
%K A014612 nonn
%O A014612 1,1
%A A014612 _Eric W. Weisstein_
%E A014612 More terms from _Patrick De Geest_, Jun 15 1998