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
Keywords
Examples
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)
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
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).
A285508 is the nonsquarefree case.
Programs
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Haskell
a014612 n = a014612_list !! (n-1) a014612_list = filter ((== 3) . a001222) [1..] -- Reinhard Zumkeller, Apr 02 2012
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Maple
with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # Wesley Ivan Hurt, Feb 05 2014
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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[244], PrimeOmega[#] == 3 &] (* Jayanta Basu, Jul 01 2013 *)
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PARI
isA014612(n)=bigomega(n)==3 \\ Charles R Greathouse IV, May 07 2011
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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
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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
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Python
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
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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)
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
Extensions
More terms from Patrick De Geest, Jun 15 1998
Comments