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A046310 Numbers that are divisible by exactly 8 primes counting multiplicity.

%I A046310 #51 Feb 16 2025 08:32:39
%S A046310 256,384,576,640,864,896,960,1296,1344,1408,1440,1600,1664,1944,2016,
%T A046310 2112,2160,2176,2240,2400,2432,2496,2916,2944,3024,3136,3168,3240,
%U A046310 3264,3360,3520,3600,3648,3712,3744,3968,4000,4160,4374,4416,4536,4704,4736
%N A046310 Numbers that are divisible by exactly 8 primes counting multiplicity.
%C A046310 Also called 8-almost primes. Products of exactly 8 primes (not necessarily distinct). Any 8-almost prime can be represented in several ways as a product of two 4-almost primes A014613 and in several ways as a product of four semiprimes A001358. - _Jonathan Vos Post_, Dec 11 2004
%C A046310 Odd terms are in A046321; first odd term is a(64)=6561=3^8. - _Zak Seidov_, Feb 08 2016
%H A046310 T. D. Noe, <a href="/A046310/b046310.txt">Table of n, a(n) for n = 1..10000</a>
%H A046310 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Reference</a>
%F A046310 Product p_i^e_i with Sum e_i = 8.
%F A046310 a(n) ~ 5040n log n / (log log n)^7. - _Charles R Greathouse IV_, May 06 2013
%F A046310 a(n) = A078840(8,n). - _R. J. Mathar_, Jan 30 2019
%p A046310 A046310 := proc(n)
%p A046310     option remember;
%p A046310     if n = 1 then
%p A046310         2^8 ;
%p A046310     else
%p A046310         for a from procname(n-1)+1 do
%p A046310             if numtheory[bigomega](a) = 8 then
%p A046310                 return a;
%p A046310             end if;
%p A046310         end do:
%p A046310     end if;
%p A046310 end proc:
%p A046310 seq(A046310(n),n=1..30) ; # _R. J. Mathar_, Dec 21 2018
%t A046310 Select[Range[1600], Plus @@ Last /@ FactorInteger[ # ] == 8 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A046310 Select[Range[5000],PrimeOmega[#]==8&]  (* _Harvey P. Dale_, Apr 19 2011 *)
%o A046310 (PARI) is(n)=bigomega(n)==8 \\ _Charles R Greathouse IV_, Mar 21 2013
%o A046310 (Python)
%o A046310 from math import prod, isqrt
%o A046310 from sympy import primerange, integer_nthroot, primepi
%o A046310 def A046310(n):
%o A046310     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 A046310     def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,8)))
%o A046310     kmin, kmax = 1,2
%o A046310     while f(kmax) >= kmax:
%o A046310         kmax <<= 1
%o A046310     while True:
%o A046310         kmid = kmax+kmin>>1
%o A046310         if f(kmid) < kmid:
%o A046310             kmax = kmid
%o A046310         else:
%o A046310             kmin = kmid
%o A046310         if kmax-kmin <= 1:
%o A046310             break
%o A046310     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A046310 Cf. A046309, A120049 (number of 8-almost primes <= 10^n).
%Y A046310 Cf. A101637, A101638, A101605, A101606.
%Y A046310 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), this sequence (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).
%Y A046310 Cf. A046321.
%K A046310 nonn
%O A046310 1,1
%A A046310 _Patrick De Geest_, Jun 15 1998