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A046310 Numbers that are divisible by exactly 8 primes counting multiplicity. 50

%I

%S 256,384,576,640,864,896,960,1296,1344,1408,1440,1600,1664,1944,2016,

%T 2112,2160,2176,2240,2400,2432,2496,2916,2944,3024,3136,3168,3240,

%U 3264,3360,3520,3600,3648,3712,3744,3968,4000,4160,4374,4416,4536,4704,4736

%N Numbers that are divisible by exactly 8 primes counting multiplicity.

%C 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 Odd terms are in A046321; first odd term is a(64)=6561=3^8. - _Zak Seidov_, Feb 08 2016

%H T. D. Noe, <a href="/A046310/b046310.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Reference</a>

%F Product p_i^e_i with Sum e_i = 8.

%F a(n) ~ 5040n log n / (log log n)^7. - _Charles R Greathouse IV_, May 06 2013

%F a(n) = A078840(8,n). - _R. J. Mathar_, Jan 30 2019

%p A046310 := proc(n)

%p option remember;

%p if n = 1 then

%p 2^8 ;

%p else

%p for a from procname(n-1)+1 do

%p if numtheory[bigomega](a) = 8 then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc:

%p seq(A046310(n),n=1..30) ; # _R. J. Mathar_, Dec 21 2018

%t Select[Range[1600], Plus @@ Last /@ FactorInteger[ # ] == 8 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)

%t Select[Range[5000],PrimeOmega[#]==8&] (* _Harvey P. Dale_, Apr 19 2011 *)

%o (PARI) is(n)=bigomega(n)==8 \\ _Charles R Greathouse IV_, Mar 21 2013

%Y Cf. A046309, A120049 (number of 8-almost primes <= 10^n).

%Y Cf. A101637, A101638, A101605, A101606.

%Y 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 Cf. A046321.

%K nonn

%O 1,1

%A _Patrick De Geest_, Jun 15 1998

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Last modified July 31 03:28 EDT 2021. Contains 346367 sequences. (Running on oeis4.)