%I #46 Dec 30 2025 11:44:01

%S 105,165,195,231,255,273,285,345,357,385,399,429,435,455,465,483,555,

%T 561,595,609,615,627,645,651,663,665,705,715,741,759,777,795,805,861,

%U 885,897,903,915,935,957,969,987,1001,1005,1015,1023,1045,1065,1085,1095,1105,1113,1131,1173

%N Products of exactly three distinct odd primes.

%C The old name was "Odd numbers with exactly 3 distinct prime factors", which is slightly ambiguous, since it might be interpreted to include 315 = 3^2*5*7. Cf. A278569. - _N. J. A. Sloane_, Nov 27 2016

%C These are the odd integers with Omega = omega = 3. - _Charles Kusniec_, Dec 27 2025

%H T. D. Noe, <a href="/A046389/b046389.txt">Table of n, a(n) for n=1..1000</a>

%p q:= n-> is([1$3]=ifactors(n)[2][.., 2]):

%p select(q, [2*i+1$i=1..590])[]; # _Alois P. Heinz_, Dec 27 2025

%t f[n_] := OddQ[n] && Last/@FactorInteger[n]=={1,1,1}; lst={}; Do[If[f[n], AppendTo[lst,n]], {n, 2000}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 23 2009 *)

%t Select[Range[1,1200,2],PrimeOmega[#]==PrimeNu[#]==3&] (* _Harvey P. Dale_, Nov 18 2025 *)

%o (PARI) list(lim)=my(v=List(), t); forprime(p=3, lim^(1/3), forprime(q=p+1, sqrt(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 26 2011

%o (Python)

%o from math import isqrt

%o from sympy import primepi, integer_nthroot, primerange

%o def A046389(n):

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))

%o return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024

%Y Intersection of A005408 and A007304.

%Y Cf. A001221, A001222, A046316, A046405, A278569.

%K nonn,easy

%O 1,1

%A _Patrick De Geest_, Jun 15 1998

%E Name clarified by _N. J. A. Sloane_, Nov 27 2016