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Numbers that are divisible by at least 10 primes (counted with multiplicity).
2

%I #22 Aug 24 2024 01:57:44

%S 1024,1536,2048,2304,2560,3072,3456,3584,3840,4096,4608,5120,5184,

%T 5376,5632,5760,6144,6400,6656,6912,7168,7680,7776,8064,8192,8448,

%U 8640,8704,8960,9216,9600,9728,9984,10240,10368,10752,11264,11520,11664,11776

%N Numbers that are divisible by at least 10 primes (counted with multiplicity).

%H John Cerkan, <a href="/A046313/b046313.txt">Table of n, a(n) for n = 1..10000</a>

%F Product p_i^e_i with Sum e_i >= 10.

%F a(n) = n + O(n (log log n)^8/log n). - _Charles R Greathouse IV_, Apr 07 2017

%t Select[Range[12000],PrimeOmega[#]>9&] (* _Harvey P. Dale_, Dec 17 2018 *)

%o (PARI) is(n)=bigomega(n)>9 \\ _Charles R Greathouse IV_, Sep 17 2015

%o (Python)

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o def A046313(n):

%o 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 def f(x): return int(n+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,10)))

%o kmin, kmax = 1,2

%o while f(kmax) >= kmax:

%o kmax <<= 1

%o while True:

%o kmid = kmax+kmin>>1

%o if f(kmid) < kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o if kmax-kmin <= 1:

%o break

%o return kmax # _Chai Wah Wu_, Aug 23 2024

%Y Subsequence of A033987, A046304, A046305, A046307, A046309, and A046311.

%Y Cf. A046314.

%K nonn

%O 1,1

%A _Patrick De Geest_, Jun 15 1998