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A046312 Numbers that are divisible by exactly 9 primes with multiplicity.

%I A046312 #41 Feb 16 2025 08:32:39
%S A046312 512,768,1152,1280,1728,1792,1920,2592,2688,2816,2880,3200,3328,3888,
%T A046312 4032,4224,4320,4352,4480,4800,4864,4992,5832,5888,6048,6272,6336,
%U A046312 6480,6528,6720,7040,7200,7296,7424,7488,7936,8000,8320,8748,8832,9072,9408
%N A046312 Numbers that are divisible by exactly 9 primes with multiplicity.
%C A046312 Also called 9-almost primes. Products of exactly 9 primes (not necessarily distinct). - _Jonathan Vos Post_, Dec 11 2004
%H A046312 T. D. Noe, <a href="/A046312/b046312.txt">Table of n, a(n) for n = 1..10000</a>
%H A046312 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Reference</a>
%F A046312 Product p_i^e_i with Sum e_i = 9.
%F A046312 a(n) ~ 40320n log n / (log log n)^8. - _Charles R Greathouse IV_, May 06 2013
%t A046312 Select[Range[2200], Plus @@ Last /@ FactorInteger[ # ] == 9 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A046312 Select[Range[10000],PrimeOmega[#]==9&] (* _Harvey P. Dale_, Oct 24 2020 *)
%o A046312 (PARI) is(n)=bigomega(n)==9 \\ _Charles R Greathouse IV_, Mar 21 2013
%o A046312 (Python)
%o A046312 from math import isqrt, prod
%o A046312 from sympy import primerange, integer_nthroot, primepi
%o A046312 def A046312(n):
%o A046312     def bisection(f,kmin=0,kmax=1):
%o A046312         while f(kmax) > kmax: kmax <<= 1
%o A046312         while kmax-kmin > 1:
%o A046312             kmid = kmax+kmin>>1
%o A046312             if f(kmid) <= kmid:
%o A046312                 kmax = kmid
%o A046312             else:
%o A046312                 kmin = kmid
%o A046312         return kmax
%o A046312     def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
%o A046312     def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,9)))
%o A046312     return bisection(f,n,n) # _Chai Wah Wu_, Nov 03 2024
%Y A046312 Cf. A046311, A120050 (number of 9-almost primes <= 10^n).
%Y A046312 Cf. A101637, A101638, A101605, A101606.
%Y A046312 Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), this sequence (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
%K A046312 nonn
%O A046312 1,1
%A A046312 _Patrick De Geest_, Jun 15 1998