1,1

Sometimes called "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

Even the first 10K terms look fairly linear; even after subtracting out the linear portion the plot looks fairly straight, although perhaps the variation is increasing with n. - Richard A. Becker, Oct 02 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 (jvospost2(AT)yahoo.com), Sep 16 2005

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.

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

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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].

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, fQ[ # ] &] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 04 2006)

Select[Range[160], Plus @@ Last /@ FactorInteger[ # ] == 3 &] - Vladimir Orlovsky, Apr 23 2008

Cf. A033942, A086062, A098238, A123072, A123073.

Cf. A109251 (number of 3-almost primes <= 10^n).

Adjacent sequences: A014609 A014610 A014611 this_sequence A014613 A014614 A014615

Sequence in context: A051144 A067537 A046339 this_sequence A046369 A066428 A054397

nonn

Eric Weisstein (eric(AT)weisstein.com)

More terms from Patrick De Geest (pdg(AT)worldofnumbers.com), Jun 15 1998.